MATH 332 - ADVANCED CALCULUS -
Differentiating Logarithmic Functions with Base
e Question Bank
Question 1
f(x) = ln(2x+ 1)
Solution:
To differentiate the function f(x) = ln(2x+ 1) with base e, we use the
formula for differentiating logarithmic functions:
d
dx ln(u) = 1
u
·
du
dx
In this case, our function is u= 2x+ 1, so we have:
d
dx ln(2x+ 1) = 1
2x+ 1
·
d
dx (2x+ 1)
Now, we find the derivative of 2x+ 1 with respect to x:
d
dx (2x+ 1) = 2
Plugging this back into our formula, we get:
d
dx ln(2x+ 1) = 1
2x+ 1
·2 = 2
2x+ 1
Therefore, the derivative of the function f(x) = ln(2x+ 1) with base eis:
d
dx ln(2x+ 1) = 2
2x+ 1
Question 1: Differentiate the following logarithmic function with base
e:
f(x) = ln(2x+ 1)
Solution:
1
To differentiate the function f(x) = ln(2x+ 1) with base e, we use
the formula for differentiating logarithmic functions:
d
dx ln(u) = 1
u
·
du
dx
In this case, our function is u= 2x+ 1, so we have:
d
dx ln(2x+ 1) = 1
2x+ 1
·
d
dx (2x+ 1)
Now, we find the derivative of 2x+ 1 with respect to x:
d
dx (2x+ 1) = 2
Plugging this back into our formula, we get:
d
dx ln(2x+ 1) = 1
2x+ 1
·2 = 2
2x+ 1
Therefore, the derivative of the function f(x) = ln(2x+ 1) with base
eis:
d
dx ln(2x+ 1) = 2
2x+ 1
Question 2
y= 5 ln(3x2
−4x)
Solution: To differentiate the given function y= 5 ln(3x2−4x)with
base e, we use the chain rule since the function is the natural loga-
rithm, which has a base of e.
Let’s denote u= 3x2−4x, so that y= 5 ln(u).
Now, we can differentiate using the chain rule, which states that
if y= ln(u), then dy
dx =1
u
du
dx .
1. Find dy
dx using the chain rule:
dy
dx = 5 ·
1
u
·
du
dx =5
3x2−4x
·(6x−4)
2. Simplify the expression:
dy
dx =5(6x−4)
3x2−4x=30x−20
3x2−4x
Therefore, the derivative of the function y= 5 ln(3x2−4x)with re-
spect to xis 30x−20
3x2−4x.Question 2: Differentiate the following logarithmic
function with base e:
2
y= 5 ln(3x2
−4x)
Solution: To differentiate the given function y= 5 ln(3x2−4x)with
base e, we use the chain rule since the function is the natural loga-
rithm, which has a base of e.
Let’s denote u= 3x2−4x, so that y= 5 ln(u).
Now, we can differentiate using the chain rule, which states that
if y= ln(u), then dy
dx =1
u
du
dx .
1. Find dy
dx using the chain rule:
dy
dx = 5 ·
1
u
·
du
dx =5
3x2−4x
·(6x−4)
2. Simplify the expression:
dy
dx =5(6x−4)
3x2−4x=30x−20
3x2−4x
Therefore, the derivative of the function y= 5 ln(3x2−4x)with
respect to xis 30x−20
3x2−4x.
Question 3
Step-by-step solution: To differentiate a logarithmic function with
base e, we use the formula d
dx (ln(u)) = 1
u
du
dx .
1. Differentiate the first term 3 ln(2x):
y1= 3 ln(2x)
dy1
dx = 3 ·
1
2x
·2
dy1
dx = 3 ·
1
x
2. Differentiate the second term 5 ln(x2):
y2= 5 ln(x2)
dy2
dx = 5 ·
1
x2
·2x
dy2
dx = 10 ·
1
x
3. Add the derivatives of the individual terms to find the overall
derivative of the function:
dy
dx =dy1
dx +dy2
dx
3
dy
dx = 3 ·
1
x+ 10 ·
1
x
dy
dx =3 + 10
x
dy
dx =13
x
Therefore, the derivative of the function y= 3 ln(2x) + 5 ln(x2)with
base eis 13
x.Question 3: Differentiate the following function with base
e:
y= 3 ln(2x) + 5 ln(x2)
Step-by-step solution: To differentiate a logarithmic function with
base e, we use the formula d
dx (ln(u)) = 1
u
du
dx .
1. Differentiate the first term 3 ln(2x):
y1= 3 ln(2x)
dy1
dx = 3 ·
1
2x
·2
dy1
dx = 3 ·
1
x
2. Differentiate the second term 5 ln(x2):
y2= 5 ln(x2)
dy2
dx = 5 ·
1
x2
·2x
dy2
dx = 10 ·
1
x
3. Add the derivatives of the individual terms to find the overall
derivative of the function:
dy
dx =dy1
dx +dy2
dx
dy
dx = 3 ·
1
x+ 10 ·
1
x
dy
dx =3 + 10
x
dy
dx =13
x
Therefore, the derivative of the function y= 3 ln(2x) + 5 ln(x2)with
base eis 13
x.
4
Question 4
Step-by-step solution: To differentiate the function f(x) = ln(2x3−
5x+ 1) with base e, we use the chain rule along with the derivative of
the natural logarithm function.
1. Let u= 2x3−5x+ 1. 2. Find the derivative of uwith respect to
x:du
dx = 6x2
−5
Now, differentiate the function with base eusing the chain rule:
d
dx [ln(u)] = 1
u
·
du
dx
d
dx [ln(2x3
−5x+ 1)] = 1
2x3−5x+ 1
·(6x2
−5)
Therefore, the derivative of f(x) = ln(2x3−5x+ 1) with base eis:
d
dx [ln(2x3
−5x+ 1)] = 6x2−5
2x3−5x+ 1
Question 4: Differentiate the following function with base e, where
f(x) = ln(2x3−5x+ 1).
Step-by-step solution: To differentiate the function f(x) = ln(2x3−
5x+ 1) with base e, we use the chain rule along with the derivative of
the natural logarithm function.
1. Let u= 2x3−5x+ 1. 2. Find the derivative of uwith respect to
x:du
dx = 6x2
−5
Now, differentiate the function with base eusing the chain rule:
d
dx [ln(u)] = 1
u
·
du
dx
d
dx [ln(2x3
−5x+ 1)] = 1
2x3−5x+ 1
·(6x2
−5)
Therefore, the derivative of f(x) = ln(2x3−5x+ 1) with base eis:
d
dx [ln(2x3
−5x+ 1)] = 6x2−5
2x3−5x+ 1
Question 5
Step-by-step solution: To differentiate the logarithmic function
with base e, we apply the chain rule.
Given function: y= ln(3x2+ 5x)
5
1. Apply the chain rule:
dy
dx =1
3x2+ 5x
·(6x+ 5)
dy
dx =6x+ 5
3x2+ 5x
Therefore, the derivative of y= ln(3x2+ 5x)with respect to xis
6x+5
3x2+5x.Question 5: Differentiate the following logarithmic function
with base e:y= ln(3x2+ 5x)
Step-by-step solution: To differentiate the logarithmic function
with base e, we apply the chain rule.
Given function: y= ln(3x2+ 5x)
1. Apply the chain rule:
dy
dx =1
3x2+ 5x
·(6x+ 5)
dy
dx =6x+ 5
3x2+ 5x
Therefore, the derivative of y= ln(3x2+ 5x)with respect to xis
6x+5
3x2+5x.
Question 6
Differentiate the following function with respect to x:
f(x) = 2 ln(e3x−5)
Solution:
To differentiate the function f(x) = 2 ln(e3x−5), we will use the
properties of logarithms and the chain rule.
1. Let’s first rewrite the function using the property ln(a) = ln(ea):
f(x) = 2 ln(e3x−5) = 2 ln(e3x−5)
2. Apply the chain rule to differentiate f(x) = 2 ln(e3x−5):
d
dx [2 ln(e3x−5)] = 2 ·
1
e3x−5
·
d
dx (e3x−5)
3. Find the derivative of e3x−5using the chain rule:
d
dx (e3x−5) = 3e3x
4. Substitute the derivative back into the expression:
2·
1
e3x−5
·3e3x
6
5. Simplify the expression:
6·
e3x
e3x−5
Therefore, the derivative of the function f(x) = 2 ln(e3x−5) is:
6e3x
e3x−5
Question 6:
Differentiate the following function with respect to x:
f(x) = 2 ln(e3x−5)
Solution:
To differentiate the function f(x) = 2 ln(e3x−5), we will use the
properties of logarithms and the chain rule.
1. Let’s first rewrite the function using the property ln(a) = ln(ea):
f(x) = 2 ln(e3x−5) = 2 ln(e3x−5)
2. Apply the chain rule to differentiate f(x) = 2 ln(e3x−5):
d
dx [2 ln(e3x−5)] = 2 ·
1
e3x−5
·
d
dx (e3x−5)
3. Find the derivative of e3x−5using the chain rule:
d
dx (e3x−5) = 3e3x
4. Substitute the derivative back into the expression:
2·
1
e3x−5
·3e3x
5. Simplify the expression:
6·
e3x
e3x−5
Therefore, the derivative of the function f(x) = 2 ln(e3x−5) is:
6e3x
e3x−5
7
Question 7
f(x) = 4 ln(3x)
Step-by-step solution: To differentiate the given function f(x) =
4 ln(3x)with base e, we will use the chain rule.
Step 1: Identify the function and its derivative. Let u= 3xand
y= ln(u).
Step 2: Find dy
du and du
dx . Since y= ln(u), we have dy
du =1
uand du
dx = 3.
Step 3: Apply the chain rule.
df
dx =df
du
·
du
dx = 4 ·
1
3x
·3=4·
1
x=4
x
Therefore, the derivative of the function f(x) = 4 ln(3x)with base e
is 4
x.Question 7: Differentiate the following logarithmic function with
base e:
f(x) = 4 ln(3x)
Step-by-step solution: To differentiate the given function f(x) =
4 ln(3x)with base e, we will use the chain rule.
Step 1: Identify the function and its derivative. Let u= 3xand
y= ln(u).
Step 2: Find dy
du and du
dx . Since y= ln(u), we have dy
du =1
uand du
dx = 3.
Step 3: Apply the chain rule.
df
dx =df
du
·
du
dx = 4 ·
1
3x
·3=4·
1
x=4
x
Therefore, the derivative of the function f(x) = 4 ln(3x)with base
eis 4
x.
Question 8
Solution: To differentiate the given function, we will apply the
product rule, which states that if f(x) = u(x)v(x), then f′(x) = u(x)v′(x)+
v(x)u′(x).
Given function: f(x)=5e3xln(2x)
Let u(x)=5e3xand v(x) = ln(2x).
Now, we differentiate u(x)and v(x)separately:
u′(x) = 5 ·3e3x= 15e3x
v′(x) = 1
2x
Now, we use the product rule to find the derivative of f(x):
f′(x) = u(x)v′(x) + v(x)u′(x)
f′(x) = (5e3x)1
2x+ ln(2x)(15e3x)
f′(x) = 5e3x
2x+ 15e3xln(2x)
8
Therefore, the derivative of the function f(x)=5e3xln(2x)with
respect to xis 5e3x
2x+15e3xln(2x).Question 8: Differentiate the function
f(x) = 5e3xln(2x)with respect to x.
Solution: To differentiate the given function, we will apply the
product rule, which states that if f(x) = u(x)v(x), then f′(x) = u(x)v′(x)+
v(x)u′(x).
Given function: f(x)=5e3xln(2x)
Let u(x)=5e3xand v(x) = ln(2x).
Now, we differentiate u(x)and v(x)separately:
u′(x) = 5 ·3e3x= 15e3x
v′(x) = 1
2x
Now, we use the product rule to find the derivative of f(x):
f′(x) = u(x)v′(x) + v(x)u′(x)
f′(x) = (5e3x)1
2x+ ln(2x)(15e3x)
f′(x) = 5e3x
2x+ 15e3xln(2x)
Therefore, the derivative of the function f(x)=5e3xln(2x)with
respect to xis 5e3x
2x+ 15e3xln(2x).
Question 9
Solution: To differentiate the given function f(x) = 3 ln(x2)with
respect to x, we will use the properties of logarithmic functions and
the chain rule.
1. Recall the derivative of the natural logarithm function: d
dx (ln(u)) =
1
u
·du
dx
2. Applying the chain rule, we have:
d
dx (3 ln(x2)) = 3 ·
1
x2
·
d
dx (x2)
3. Simplifying the expression, we get:
d
dx (3 ln(x2)) = 3 ·
1
x2
·2x
4. Further simplifying, we have:
d
dx (3 ln(x2)) = 6 ·
1
x
5. Therefore, the derivative of the function f(x) = 3 ln(x2)with
respect to xis:
d
dx (3 ln(x2)) = 6
x
Question 9: Differentiate the following function with respect to x:
f(x) = 3 ln(x2)
9
Solution: To differentiate the given function f(x) = 3 ln(x2)with
respect to x, we will use the properties of logarithmic functions and
the chain rule.
1. Recall the derivative of the natural logarithm function: d
dx (ln(u)) =
1
u
·du
dx
2. Applying the chain rule, we have:
d
dx (3 ln(x2)) = 3 ·
1
x2
·
d
dx (x2)
3. Simplifying the expression, we get:
d
dx (3 ln(x2)) = 3 ·
1
x2
·2x
4. Further simplifying, we have:
d
dx (3 ln(x2)) = 6 ·
1
x
5. Therefore, the derivative of the function f(x) = 3 ln(x2)with
respect to xis:
d
dx (3 ln(x2)) = 6
x
Question 10
Step-by-step solution: To differentiate the function f(x) = ln(2x2),
we use the fact that the derivative of ln(u)is 1
u
·u′, where uis a function
of x.
1. Apply the differentiation rule:
f′(x) = 1
2x2
·(2x)
f′(x) = 2x
2x2
f′(x) = 1
x
Therefore, the derivative of f(x) = ln(2x2)with respect to xis
f′(x) = 1
x.Question 10: Differentiate the following logarithmic function
with base e:
f(x) = ln(2x2)
Step-by-step solution: To differentiate the function f(x) = ln(2x2),
we use the fact that the derivative of ln(u)is 1
u
·u′, where uis a function
of x.
10
1. Apply the differentiation rule:
f′(x) = 1
2x2
·(2x)
f′(x) = 2x
2x2
f′(x) = 1
x
Therefore, the derivative of f(x) = ln(2x2)with respect to xis
f′(x) = 1
x.
Question 11
Step-by-Step Solution: Given function: f(x) = 3 ln(2x)
Applying the properties of logarithms: f(x) = 3 ln(2) + 3 ln(x)
Now, differentiate with respect to xusing the derivative of natural
logarithm: d
dx [3 ln(2) + 3 ln(x)]
Using the derivative of a constant times a function: d
dx [3 ln(2)] +
d
dx [3 ln(x)]
Since ln(2) is a constant, the derivative of a constant is 0: 0+3·
1
x
Simplify the expression: 3
x
Therefore, the derivative of the function f(x) = 3 ln(2x)is 3
x.Question
11: Find the derivative of the function f(x) = 3 ln(2x).
Step-by-Step Solution: Given function: f(x) = 3 ln(2x)
Applying the properties of logarithms: f(x) = 3 ln(2) + 3 ln(x)
Now, differentiate with respect to xusing the derivative of natural
logarithm: d
dx [3 ln(2) + 3 ln(x)]
Using the derivative of a constant times a function: d
dx [3 ln(2)] +
d
dx [3 ln(x)]
Since ln(2) is a constant, the derivative of a constant is 0: 0+3·
1
x
Simplify the expression: 3
x
Therefore, the derivative of the function f(x) = 3 ln(2x)is 3
x.
Question 12
Solution: To differentiate the function f(x) = ln(5x3), we use the
chain rule, which states that if we have a composition of functions
h(g(x)), then the derivative is given by (h′(g(x))) ·g′(x).
Given function: f(x) = ln(5x3).
Step 1: Identify the inner function g(x) = 5x3and the outer func-
tion h(x) = ln(x).
Step 2: Find the derivative of the inner function:
g′(x)=3·5x3−1= 15x2
11
Step 3: Find the derivative of the outer function:
h′(x) = 1
x
Step 4: Apply the chain rule:
f′(x) = h′(g(x)) ·g′(x) = 1
5x3
·15x2
Step 5: Simplify the expression:
f′(x) = 15x2
5x3=3
x
Therefore, the derivative of the function f(x) = ln(5x3)is f′(x) =
3
x.Question 12: Differentiate the function f(x) = ln(5x3).
Solution: To differentiate the function f(x) = ln(5x3), we use the
chain rule, which states that if we have a composition of functions
h(g(x)), then the derivative is given by (h′(g(x))) ·g′(x).
Given function: f(x) = ln(5x3).
Step 1: Identify the inner function g(x) = 5x3and the outer func-
tion h(x) = ln(x).
Step 2: Find the derivative of the inner function:
g′(x)=3·5x3−1= 15x2
Step 3: Find the derivative of the outer function:
h′(x) = 1
x
Step 4: Apply the chain rule:
f′(x) = h′(g(x)) ·g′(x) = 1
5x3
·15x2
Step 5: Simplify the expression:
f′(x) = 15x2
5x3=3
x
Therefore, the derivative of the function f(x) = ln(5x3)is f′(x) = 3
x.
Question 13
Step-by-step Solution: To differentiate the given function f(x) =
ln(3x2+4x+1) with respect to x, we use the chain rule for differentiation
and the derivative of the natural logarithm function.
12
1. Use the chain rule:
f′(x) = 1
3x2+ 4x+ 1
·
d
dx (3x2+ 4x+ 1)
2. Differentiate the inside function:
d
dx (3x2+ 4x+ 1) = 6x+ 4
3. Substitute the derivative of the inside function back into the
chain rule equation:
f′(x) = 1
3x2+ 4x+ 1
·(6x+ 4)
4. Simplify the expression:
f′(x) = 6x+ 4
3x2+ 4x+ 1
Therefore, the derivative of the function f(x) = ln(3x2+ 4x+ 1)
with respect to xis 6x+4
3x2+4x+1 .Question 13: Differentiate the following
function with respect to x:
f(x) = ln(3x2+ 4x+ 1)
Step-by-step Solution: To differentiate the given function f(x) =
ln(3x2+4x+1) with respect to x, we use the chain rule for differentiation
and the derivative of the natural logarithm function.
1. Use the chain rule:
f′(x) = 1
3x2+ 4x+ 1
·
d
dx (3x2+ 4x+ 1)
2. Differentiate the inside function:
d
dx (3x2+ 4x+ 1) = 6x+ 4
3. Substitute the derivative of the inside function back into the
chain rule equation:
f′(x) = 1
3x2+ 4x+ 1
·(6x+ 4)
4. Simplify the expression:
f′(x) = 6x+ 4
3x2+ 4x+ 1
Therefore, the derivative of the function f(x) = ln(3x2+ 4x+ 1) with
respect to xis 6x+4
3x2+4x+1 .
13
Question 14
Step-by-step solution: To differentiate the function f(x) = ln(3x2−
4x), we will use the chain rule: 1. Let u= 3x2−4x. 2. Apply the
chain rule: d
dx (ln(u)) = 1
u
·du
dx . 3. Find the derivative of u:du
dx = 6x−4.
4. Substitute the derivative of uback into the chain rule formula:
d
dx (ln(3x2−4x)) = 1
3x2−4x
·(6x−4). 5. Simplify the expression: f′(x) =
6x−4
3x2−4x.
Therefore, the derivative of the function f(x) = ln(3x2−4x)with
respect to xis f′(x) = 6x−4
3x2−4x.Question 14: Differentiate the following
logarithmic function with base e:f(x) = ln(3x2−4x)
Step-by-step solution: To differentiate the function f(x) = ln(3x2−
4x), we will use the chain rule: 1. Let u= 3x2−4x. 2. Apply the
chain rule: d
dx (ln(u)) = 1
u
·du
dx . 3. Find the derivative of u:du
dx = 6x−4.
4. Substitute the derivative of uback into the chain rule formula:
d
dx (ln(3x2−4x)) = 1
3x2−4x
·(6x−4). 5. Simplify the expression: f′(x) =
6x−4
3x2−4x.
Therefore, the derivative of the function f(x) = ln(3x2−4x)with
respect to xis f′(x) = 6x−4
3x2−4x.
Question 15
Question 15: Differentiate the following function with respect to
x:
f(x) = ln(3x2
−5x+ 2)
Solution: To differentiate the given function f(x) = ln(3x2−5x+ 2)
with respect to x, we will use the chain rule of differentiation.
Step 1: Let u= 3x2−5x+ 2.
Step 2: Differentiate uwith respect to xto get du
dx .
du
dx = 6x−5
Step 3: Apply the chain rule to differentiate the function f(x) =
ln(u).
df
dx =1
u
·
du
dx
Substitute u= 3x2−5x+ 2 and du
dx = 6x−5into the formula:
df
dx =1
3x2−5x+ 2
·(6x−5)
Therefore, the derivative of the function f(x) = ln(3x2−5x+ 2) with
respect to xis:
14
df
dx =6x−5
3x2−5x+ 2
This is the final answer.Sure! Here is the question along with its
step-by-step solution in LateX code:
Question 15: Differentiate the following function with respect to
x:
f(x) = ln(3x2
−5x+ 2)
Solution: To differentiate the given function f(x) = ln(3x2−5x+ 2)
with respect to x, we will use the chain rule of differentiation.
Step 1: Let u= 3x2−5x+ 2.
Step 2: Differentiate uwith respect to xto get du
dx .
du
dx = 6x−5
Step 3: Apply the chain rule to differentiate the function f(x) =
ln(u).
df
dx =1
u
·
du
dx
Substitute u= 3x2−5x+ 2 and du
dx = 6x−5into the formula:
df
dx =1
3x2−5x+ 2
·(6x−5)
Therefore, the derivative of the function f(x) = ln(3x2−5x+ 2) with
respect to xis:
df
dx =6x−5
3x2−5x+ 2
This is the final answer.
Question 16
y= 3 ln(x)−2 ln(x2)
Step-by-step solution: To differentiate the given function, we will
apply the properties of logarithms and the chain rule of differentia-
tion.
1. Given Function:
y= 3 ln(x)−2 ln(x2)
15
2. Apply the properties of logarithms:
y= ln(x3)−ln(x4)
y= ln(x3
x4)
y= ln(x−1)
3. Rewrite the function:
y=−ln(x)
4. Differentiate the function using the differentiation rule for ln(x):
dy
dx =−
1
x
5. Hence, the derivative of the given logarithmic function with
base eis: dy
dx =−
1
x
Therefore, the derivative of 3 ln(x)−2 ln(x2)with respect to xis
−
1
x.Question 16: Differentiate the following logarithmic function with
base e:
y= 3 ln(x)−2 ln(x2)
Step-by-step solution: To differentiate the given function, we will
apply the properties of logarithms and the chain rule of differentia-
tion.
1. Given Function:
y= 3 ln(x)−2 ln(x2)
2. Apply the properties of logarithms:
y= ln(x3)−ln(x4)
y= ln(x3
x4)
y= ln(x−1)
3. Rewrite the function:
y=−ln(x)
4. Differentiate the function using the differentiation rule for ln(x):
dy
dx =−
1
x
5. Hence, the derivative of the given logarithmic function with
base eis: dy
dx =−
1
x
Therefore, the derivative of 3 ln(x)−2 ln(x2)with respect to xis −
1
x.
16
Question 17
Differentiate the following function with respect to x:f(x) = 3 ln(4x2−
5).
Solution:
To differentiate the given function f(x) = 3 ln(4x2−5) with respect
to x, we will use the rule for differentiating logarithmic functions with
base e.
1. Let’s rewrite the function in a different form to make the differ-
entiation process easier: f(x) = 3 ln(4x2−5) = 3 ln(u), where u= 4x2−5.
2. Now, differentiate the function f(x) = 3 ln(u)using the chain
rule:
d
dx [3 ln(u)] = 3 ·
d
dx [ln(u)]
= 3 ·
1
u
·
du
dx
= 3 ·
1
4x2−5
·8x
=24x
4x2−5
3. Therefore, the derivative of the function f(x) = 3 ln(4x2−5) with
respect to xis:
d
dx 3 ln(4x2
−5)=24x
4x2−5
Question 17:
Differentiate the following function with respect to x:f(x) = 3 ln(4x2−
5).
Solution:
To differentiate the given function f(x) = 3 ln(4x2−5) with respect
to x, we will use the rule for differentiating logarithmic functions with
base e.
1. Let’s rewrite the function in a different form to make the differ-
entiation process easier: f(x) = 3 ln(4x2−5) = 3 ln(u), where u= 4x2−5.
2. Now, differentiate the function f(x) = 3 ln(u)using the chain
rule:
d
dx [3 ln(u)] = 3 ·
d
dx [ln(u)]
= 3 ·
1
u
·
du
dx
= 3 ·
1
4x2−5
·8x
=24x
4x2−5
17
3. Therefore, the derivative of the function f(x) = 3 ln(4x2−5) with
respect to xis:
d
dx 3 ln(4x2
−5)=24x
4x2−5
Question 18
Step-by-step solution: To differentiate the given function f(x) =
3 ln(x2+ 1), we will apply the rules of differentiation for logarithmic
functions with base e(natural logarithm).
1. Use the chain rule: differentiate the outer function (ln) first,
then multiply by the derivative of the inner function (x2+ 1).
2. The derivative of ln(u)with respect to xis 1
u
·du
dx .
3. Let u=x2+ 1. Then, du
dx = 2x.
4. Plug these values into the chain rule formula.
5. We get:
f′(x)=3·
1
x2+ 1
·2x
f′(x)=3·
2x
x2+ 1
f′(x) = 6x
x2+ 1
Therefore, the derivative of the function f(x) = 3 ln(x2+1) is 6x
x2+1 .Question
18: Differentiate the function f(x) = 3 ln(x2+ 1).
Step-by-step solution: To differentiate the given function f(x) =
3 ln(x2+ 1), we will apply the rules of differentiation for logarithmic
functions with base e(natural logarithm).
1. Use the chain rule: differentiate the outer function (ln) first,
then multiply by the derivative of the inner function (x2+ 1).
2. The derivative of ln(u)with respect to xis 1
u
·du
dx .
3. Let u=x2+ 1. Then, du
dx = 2x.
4. Plug these values into the chain rule formula.
5. We get:
f′(x)=3·
1
x2+ 1
·2x
f′(x)=3·
2x
x2+ 1
f′(x) = 6x
x2+ 1
Therefore, the derivative of the function f(x) = 3 ln(x2+ 1) is 6x
x2+1 .
18
Question 19
Differentiate the following function with respect to x, where f(x) =
3 ln(2x3+ 5x).
Solution:
Given function: f(x) = 3 ln(2x3+ 5x).
To differentiate a logarithmic function with base e, we use the
chain rule. The derivative of ln(u)with respect to xis 1
u
·du
dx .
Let u= 2x3+ 5x.
Now, we can differentiate f(x)using the chain rule:
df
dx = 3 ·
1
2x3+ 5x
·(6x2+ 5)
df
dx =3(6x2+ 5)
2x3+ 5x
Therefore, the derivative of f(x) = 3 ln(2x3+ 5x)with respect to x
is 18x2+15
2x3+5x.Question 19:
Differentiate the following function with respect to x, where f(x) =
3 ln(2x3+ 5x).
Solution:
Given function: f(x) = 3 ln(2x3+ 5x).
To differentiate a logarithmic function with base e, we use the
chain rule. The derivative of ln(u)with respect to xis 1
u
·du
dx .
Let u= 2x3+ 5x.
Now, we can differentiate f(x)using the chain rule:
df
dx = 3 ·
1
2x3+ 5x
·(6x2+ 5)
df
dx =3(6x2+ 5)
2x3+ 5x
Therefore, the derivative of f(x) = 3 ln(2x3+ 5x)with respect to x
is 18x2+15
2x3+5x.
Question 20
Solution: To differentiate the given function using logarithmic dif-
ferentiation, we follow these steps:
1. Let y= ln(3x2−2x).
2. Take the natural logarithm of both sides: ln(y) = ln(ln(3x2−2x)).
19
3. Differentiate both sides with respect to x:
d
dx [ln(y)] = d
dx [ln(3x2
−2x)]
1
y
dy
dx =1
3x2−2x
·(6x−2)
dy
dx =y·
6x−2
3x2−2x
= (ln(3x2
−2x)) ·
6x−2
3x2−2x.
4. Substitute back y= ln(3x2−2x)to get the final answer:
f′(x) = ln(3x2
−2x)·
6x−2
3x2−2x.
Question 20: Differentiate the following function with respect to
xusing logarithmic differentiation: f(x) = ln(3x2−2x).
Solution: To differentiate the given function using logarithmic dif-
ferentiation, we follow these steps:
1. Let y= ln(3x2−2x).
2. Take the natural logarithm of both sides: ln(y) = ln(ln(3x2−2x)).
3. Differentiate both sides with respect to x:
d
dx [ln(y)] = d
dx [ln(3x2
−2x)]
1
y
dy
dx =1
3x2−2x
·(6x−2)
dy
dx =y·
6x−2
3x2−2x
= (ln(3x2
−2x)) ·
6x−2
3x2−2x.
4. Substitute back y= ln(3x2−2x)to get the final answer:
f′(x) = ln(3x2
−2x)·
6x−2
3x2−2x.
20
To differentiate the function f(x) = ln(2x+ 1) with base e, we use
the formula for differentiating logarithmic functions:
d
dx ln(u) = 1
u
·
du
dx
In this case, our function is u= 2x+ 1, so we have:
d
dx ln(2x+ 1) = 1
2x+ 1
·
d
dx (2x+ 1)
Now, we find the derivative of 2x+ 1 with respect to x:
d
dx (2x+ 1) = 2
Plugging this back into our formula, we get:
d
dx ln(2x+ 1) = 1
2x+ 1
·2 = 2
2x+ 1
Therefore, the derivative of the function f(x) = ln(2x+ 1) with base
eis:
d
dx ln(2x+ 1) = 2
2x+ 1
Question 2
y= 5 ln(3x2
−4x)
Solution: To differentiate the given function y= 5 ln(3x2−4x)with
base e, we use the chain rule since the function is the natural loga-
rithm, which has a base of e.
Let’s denote u= 3x2−4x, so that y= 5 ln(u).
Now, we can differentiate using the chain rule, which states that
if y= ln(u), then dy
dx =1
u
du
dx .
1. Find dy
dx using the chain rule:
dy
dx = 5 ·
1
u
·
du
dx =5
3x2−4x
·(6x−4)
2. Simplify the expression:
dy
dx =5(6x−4)
3x2−4x=30x−20
3x2−4x
Therefore, the derivative of the function y= 5 ln(3x2−4x)with re-
spect to xis 30x−20
3x2−4x.Question 2: Differentiate the following logarithmic
function with base e:
2
y= 5 ln(3x2
−4x)
Solution: To differentiate the given function y= 5 ln(3x2−4x)with
base e, we use the chain rule since the function is the natural loga-
rithm, which has a base of e.
Let’s denote u= 3x2−4x, so that y= 5 ln(u).
Now, we can differentiate using the chain rule, which states that
if y= ln(u), then dy
dx =1
u
du
dx .
1. Find dy
dx using the chain rule:
dy
dx = 5 ·
1
u
·
du
dx =5
3x2−4x
·(6x−4)
2. Simplify the expression:
dy
dx =5(6x−4)
3x2−4x=30x−20
3x2−4x
Therefore, the derivative of the function y= 5 ln(3x2−4x)with
respect to xis 30x−20
3x2−4x.
Question 3
Step-by-step solution: To differentiate a logarithmic function with
base e, we use the formula d
dx (ln(u)) = 1
u
du
dx .
1. Differentiate the first term 3 ln(2x):
y1= 3 ln(2x)
dy1
dx = 3 ·
1
2x
·2
dy1
dx = 3 ·
1
x
2. Differentiate the second term 5 ln(x2):
y2= 5 ln(x2)
dy2
dx = 5 ·
1
x2
·2x
dy2
dx = 10 ·
1
x
3. Add the derivatives of the individual terms to find the overall
derivative of the function:
dy
dx =dy1
dx +dy2
dx
3
dy
dx = 3 ·
1
x+ 10 ·
1
x
dy
dx =3 + 10
x
dy
dx =13
x
Therefore, the derivative of the function y= 3 ln(2x) + 5 ln(x2)with
base eis 13
x.Question 3: Differentiate the following function with base
e:
y= 3 ln(2x) + 5 ln(x2)
Step-by-step solution: To differentiate a logarithmic function with
base e, we use the formula d
dx (ln(u)) = 1
u
du
dx .
1. Differentiate the first term 3 ln(2x):
y1= 3 ln(2x)
dy1
dx = 3 ·
1
2x
·2
dy1
dx = 3 ·
1
x
2. Differentiate the second term 5 ln(x2):
y2= 5 ln(x2)
dy2
dx = 5 ·
1
x2
·2x
dy2
dx = 10 ·
1
x
3. Add the derivatives of the individual terms to find the overall
derivative of the function:
dy
dx =dy1
dx +dy2
dx
dy
dx = 3 ·
1
x+ 10 ·
1
x
dy
dx =3 + 10
x
dy
dx =13
x
Therefore, the derivative of the function y= 3 ln(2x) + 5 ln(x2)with
base eis 13
x.
4
Question 4
Step-by-step solution: To differentiate the function f(x) = ln(2x3−
5x+ 1) with base e, we use the chain rule along with the derivative of
the natural logarithm function.
1. Let u= 2x3−5x+ 1. 2. Find the derivative of uwith respect to
x:du
dx = 6x2
−5
Now, differentiate the function with base eusing the chain rule:
d
dx [ln(u)] = 1
u
·
du
dx
d
dx [ln(2x3
−5x+ 1)] = 1
2x3−5x+ 1
·(6x2
−5)
Therefore, the derivative of f(x) = ln(2x3−5x+ 1) with base eis:
d
dx [ln(2x3
−5x+ 1)] = 6x2−5
2x3−5x+ 1
Question 4: Differentiate the following function with base e, where
f(x) = ln(2x3−5x+ 1).
Step-by-step solution: To differentiate the function f(x) = ln(2x3−
5x+ 1) with base e, we use the chain rule along with the derivative of
the natural logarithm function.
1. Let u= 2x3−5x+ 1. 2. Find the derivative of uwith respect to
x:du
dx = 6x2
−5
Now, differentiate the function with base eusing the chain rule:
d
dx [ln(u)] = 1
u
·
du
dx
d
dx [ln(2x3
−5x+ 1)] = 1
2x3−5x+ 1
·(6x2
−5)
Therefore, the derivative of f(x) = ln(2x3−5x+ 1) with base eis:
d
dx [ln(2x3
−5x+ 1)] = 6x2−5
2x3−5x+ 1
Question 5
Step-by-step solution: To differentiate the logarithmic function
with base e, we apply the chain rule.
Given function: y= ln(3x2+ 5x)
5
1. Apply the chain rule:
dy
dx =1
3x2+ 5x
·(6x+ 5)
dy
dx =6x+ 5
3x2+ 5x
Therefore, the derivative of y= ln(3x2+ 5x)with respect to xis
6x+5
3x2+5x.Question 5: Differentiate the following logarithmic function
with base e:y= ln(3x2+ 5x)
Step-by-step solution: To differentiate the logarithmic function
with base e, we apply the chain rule.
Given function: y= ln(3x2+ 5x)
1. Apply the chain rule:
dy
dx =1
3x2+ 5x
·(6x+ 5)
dy
dx =6x+ 5
3x2+ 5x
Therefore, the derivative of y= ln(3x2+ 5x)with respect to xis
6x+5
3x2+5x.
Question 6
Differentiate the following function with respect to x:
f(x) = 2 ln(e3x−5)
Solution:
To differentiate the function f(x) = 2 ln(e3x−5), we will use the
properties of logarithms and the chain rule.
1. Let’s first rewrite the function using the property ln(a) = ln(ea):
f(x) = 2 ln(e3x−5) = 2 ln(e3x−5)
2. Apply the chain rule to differentiate f(x) = 2 ln(e3x−5):
d
dx [2 ln(e3x−5)] = 2 ·
1
e3x−5
·
d
dx (e3x−5)
3. Find the derivative of e3x−5using the chain rule:
d
dx (e3x−5) = 3e3x
4. Substitute the derivative back into the expression:
2·
1
e3x−5
·3e3x
6
5. Simplify the expression:
6·
e3x
e3x−5
Therefore, the derivative of the function f(x) = 2 ln(e3x−5) is:
6e3x
e3x−5
Question 6:
Differentiate the following function with respect to x:
f(x) = 2 ln(e3x−5)
Solution:
To differentiate the function f(x) = 2 ln(e3x−5), we will use the
properties of logarithms and the chain rule.
1. Let’s first rewrite the function using the property ln(a) = ln(ea):
f(x) = 2 ln(e3x−5) = 2 ln(e3x−5)
2. Apply the chain rule to differentiate f(x) = 2 ln(e3x−5):
d
dx [2 ln(e3x−5)] = 2 ·
1
e3x−5
·
d
dx (e3x−5)
3. Find the derivative of e3x−5using the chain rule:
d
dx (e3x−5) = 3e3x
4. Substitute the derivative back into the expression:
2·
1
e3x−5
·3e3x
5. Simplify the expression:
6·
e3x
e3x−5
Therefore, the derivative of the function f(x) = 2 ln(e3x−5) is:
6e3x
e3x−5
7
Question 7
f(x) = 4 ln(3x)
Step-by-step solution: To differentiate the given function f(x) =
4 ln(3x)with base e, we will use the chain rule.
Step 1: Identify the function and its derivative. Let u= 3xand
y= ln(u).
Step 2: Find dy
du and du
dx . Since y= ln(u), we have dy
du =1
uand du
dx = 3.
Step 3: Apply the chain rule.
df
dx =df
du
·
du
dx = 4 ·
1
3x
·3=4·
1
x=4
x
Therefore, the derivative of the function f(x) = 4 ln(3x)with base e
is 4
x.Question 7: Differentiate the following logarithmic function with
base e:
f(x) = 4 ln(3x)
Step-by-step solution: To differentiate the given function f(x) =
4 ln(3x)with base e, we will use the chain rule.
Step 1: Identify the function and its derivative. Let u= 3xand
y= ln(u).
Step 2: Find dy
du and du
dx . Since y= ln(u), we have dy
du =1
uand du
dx = 3.
Step 3: Apply the chain rule.
df
dx =df
du
·
du
dx = 4 ·
1
3x
·3=4·
1
x=4
x
Therefore, the derivative of the function f(x) = 4 ln(3x)with base
eis 4
x.
Question 8
Solution: To differentiate the given function, we will apply the
product rule, which states that if f(x) = u(x)v(x), then f′(x) = u(x)v′(x)+
v(x)u′(x).
Given function: f(x)=5e3xln(2x)
Let u(x)=5e3xand v(x) = ln(2x).
Now, we differentiate u(x)and v(x)separately:
u′(x) = 5 ·3e3x= 15e3x
v′(x) = 1
2x
Now, we use the product rule to find the derivative of f(x):
f′(x) = u(x)v′(x) + v(x)u′(x)
f′(x) = (5e3x)1
2x+ ln(2x)(15e3x)
f′(x) = 5e3x
2x+ 15e3xln(2x)
8
Therefore, the derivative of the function f(x)=5e3xln(2x)with
respect to xis 5e3x
2x+15e3xln(2x).Question 8: Differentiate the function
f(x) = 5e3xln(2x)with respect to x.
Solution: To differentiate the given function, we will apply the
product rule, which states that if f(x) = u(x)v(x), then f′(x) = u(x)v′(x)+
v(x)u′(x).
Given function: f(x)=5e3xln(2x)
Let u(x)=5e3xand v(x) = ln(2x).
Now, we differentiate u(x)and v(x)separately:
u′(x) = 5 ·3e3x= 15e3x
v′(x) = 1
2x
Now, we use the product rule to find the derivative of f(x):
f′(x) = u(x)v′(x) + v(x)u′(x)
f′(x) = (5e3x)1
2x+ ln(2x)(15e3x)
f′(x) = 5e3x
2x+ 15e3xln(2x)
Therefore, the derivative of the function f(x)=5e3xln(2x)with
respect to xis 5e3x
2x+ 15e3xln(2x).
Question 9
Solution: To differentiate the given function f(x) = 3 ln(x2)with
respect to x, we will use the properties of logarithmic functions and
the chain rule.
1. Recall the derivative of the natural logarithm function: d
dx (ln(u)) =
1
u
·du
dx
2. Applying the chain rule, we have:
d
dx (3 ln(x2)) = 3 ·
1
x2
·
d
dx (x2)
3. Simplifying the expression, we get:
d
dx (3 ln(x2)) = 3 ·
1
x2
·2x
4. Further simplifying, we have:
d
dx (3 ln(x2)) = 6 ·
1
x
5. Therefore, the derivative of the function f(x) = 3 ln(x2)with
respect to xis:
d
dx (3 ln(x2)) = 6
x
Question 9: Differentiate the following function with respect to x:
f(x) = 3 ln(x2)
9
Solution: To differentiate the given function f(x) = 3 ln(x2)with
respect to x, we will use the properties of logarithmic functions and
the chain rule.
1. Recall the derivative of the natural logarithm function: d
dx (ln(u)) =
1
u
·du
dx
2. Applying the chain rule, we have:
d
dx (3 ln(x2)) = 3 ·
1
x2
·
d
dx (x2)
3. Simplifying the expression, we get:
d
dx (3 ln(x2)) = 3 ·
1
x2
·2x
4. Further simplifying, we have:
d
dx (3 ln(x2)) = 6 ·
1
x
5. Therefore, the derivative of the function f(x) = 3 ln(x2)with
respect to xis:
d
dx (3 ln(x2)) = 6
x
Question 10
Step-by-step solution: To differentiate the function f(x) = ln(2x2),
we use the fact that the derivative of ln(u)is 1
u
·u′, where uis a function
of x.
1. Apply the differentiation rule:
f′(x) = 1
2x2
·(2x)
f′(x) = 2x
2x2
f′(x) = 1
x
Therefore, the derivative of f(x) = ln(2x2)with respect to xis
f′(x) = 1
x.Question 10: Differentiate the following logarithmic function
with base e:
f(x) = ln(2x2)
Step-by-step solution: To differentiate the function f(x) = ln(2x2),
we use the fact that the derivative of ln(u)is 1
u
·u′, where uis a function
of x.
10
1. Apply the differentiation rule:
f′(x) = 1
2x2
·(2x)
f′(x) = 2x
2x2
f′(x) = 1
x
Therefore, the derivative of f(x) = ln(2x2)with respect to xis
f′(x) = 1
x.
Question 11
Step-by-Step Solution: Given function: f(x) = 3 ln(2x)
Applying the properties of logarithms: f(x) = 3 ln(2) + 3 ln(x)
Now, differentiate with respect to xusing the derivative of natural
logarithm: d
dx [3 ln(2) + 3 ln(x)]
Using the derivative of a constant times a function: d
dx [3 ln(2)] +
d
dx [3 ln(x)]
Since ln(2) is a constant, the derivative of a constant is 0: 0+3·
1
x
Simplify the expression: 3
x
Therefore, the derivative of the function f(x) = 3 ln(2x)is 3
x.Question
11: Find the derivative of the function f(x) = 3 ln(2x).
Step-by-Step Solution: Given function: f(x) = 3 ln(2x)
Applying the properties of logarithms: f(x) = 3 ln(2) + 3 ln(x)
Now, differentiate with respect to xusing the derivative of natural
logarithm: d
dx [3 ln(2) + 3 ln(x)]
Using the derivative of a constant times a function: d
dx [3 ln(2)] +
d
dx [3 ln(x)]
Since ln(2) is a constant, the derivative of a constant is 0: 0+3·
1
x
Simplify the expression: 3
x
Therefore, the derivative of the function f(x) = 3 ln(2x)is 3
x.
Question 12
Solution: To differentiate the function f(x) = ln(5x3), we use the
chain rule, which states that if we have a composition of functions
h(g(x)), then the derivative is given by (h′(g(x))) ·g′(x).
Given function: f(x) = ln(5x3).
Step 1: Identify the inner function g(x) = 5x3and the outer func-
tion h(x) = ln(x).
Step 2: Find the derivative of the inner function:
g′(x)=3·5x3−1= 15x2
11
Step 3: Find the derivative of the outer function:
h′(x) = 1
x
Step 4: Apply the chain rule:
f′(x) = h′(g(x)) ·g′(x) = 1
5x3
·15x2
Step 5: Simplify the expression:
f′(x) = 15x2
5x3=3
x
Therefore, the derivative of the function f(x) = ln(5x3)is f′(x) =
3
x.Question 12: Differentiate the function f(x) = ln(5x3).
Solution: To differentiate the function f(x) = ln(5x3), we use the
chain rule, which states that if we have a composition of functions
h(g(x)), then the derivative is given by (h′(g(x))) ·g′(x).
Given function: f(x) = ln(5x3).
Step 1: Identify the inner function g(x) = 5x3and the outer func-
tion h(x) = ln(x).
Step 2: Find the derivative of the inner function:
g′(x)=3·5x3−1= 15x2
Step 3: Find the derivative of the outer function:
h′(x) = 1
x
Step 4: Apply the chain rule:
f′(x) = h′(g(x)) ·g′(x) = 1
5x3
·15x2
Step 5: Simplify the expression:
f′(x) = 15x2
5x3=3
x
Therefore, the derivative of the function f(x) = ln(5x3)is f′(x) = 3
x.
Question 13
Step-by-step Solution: To differentiate the given function f(x) =
ln(3x2+4x+1) with respect to x, we use the chain rule for differentiation
and the derivative of the natural logarithm function.
12
1. Use the chain rule:
f′(x) = 1
3x2+ 4x+ 1
·
d
dx (3x2+ 4x+ 1)
2. Differentiate the inside function:
d
dx (3x2+ 4x+ 1) = 6x+ 4
3. Substitute the derivative of the inside function back into the
chain rule equation:
f′(x) = 1
3x2+ 4x+ 1
·(6x+ 4)
4. Simplify the expression:
f′(x) = 6x+ 4
3x2+ 4x+ 1
Therefore, the derivative of the function f(x) = ln(3x2+ 4x+ 1)
with respect to xis 6x+4
3x2+4x+1 .Question 13: Differentiate the following
function with respect to x:
f(x) = ln(3x2+ 4x+ 1)
Step-by-step Solution: To differentiate the given function f(x) =
ln(3x2+4x+1) with respect to x, we use the chain rule for differentiation
and the derivative of the natural logarithm function.
1. Use the chain rule:
f′(x) = 1
3x2+ 4x+ 1
·
d
dx (3x2+ 4x+ 1)
2. Differentiate the inside function:
d
dx (3x2+ 4x+ 1) = 6x+ 4
3. Substitute the derivative of the inside function back into the
chain rule equation:
f′(x) = 1
3x2+ 4x+ 1
·(6x+ 4)
4. Simplify the expression:
f′(x) = 6x+ 4
3x2+ 4x+ 1
Therefore, the derivative of the function f(x) = ln(3x2+ 4x+ 1) with
respect to xis 6x+4
3x2+4x+1 .
13
Question 14
Step-by-step solution: To differentiate the function f(x) = ln(3x2−
4x), we will use the chain rule: 1. Let u= 3x2−4x. 2. Apply the
chain rule: d
dx (ln(u)) = 1
u
·du
dx . 3. Find the derivative of u:du
dx = 6x−4.
4. Substitute the derivative of uback into the chain rule formula:
d
dx (ln(3x2−4x)) = 1
3x2−4x
·(6x−4). 5. Simplify the expression: f′(x) =
6x−4
3x2−4x.
Therefore, the derivative of the function f(x) = ln(3x2−4x)with
respect to xis f′(x) = 6x−4
3x2−4x.Question 14: Differentiate the following
logarithmic function with base e:f(x) = ln(3x2−4x)
Step-by-step solution: To differentiate the function f(x) = ln(3x2−
4x), we will use the chain rule: 1. Let u= 3x2−4x. 2. Apply the
chain rule: d
dx (ln(u)) = 1
u
·du
dx . 3. Find the derivative of u:du
dx = 6x−4.
4. Substitute the derivative of uback into the chain rule formula:
d
dx (ln(3x2−4x)) = 1
3x2−4x
·(6x−4). 5. Simplify the expression: f′(x) =
6x−4
3x2−4x.
Therefore, the derivative of the function f(x) = ln(3x2−4x)with
respect to xis f′(x) = 6x−4
3x2−4x.
Question 15
Question 15: Differentiate the following function with respect to
x:
f(x) = ln(3x2
−5x+ 2)
Solution: To differentiate the given function f(x) = ln(3x2−5x+ 2)
with respect to x, we will use the chain rule of differentiation.
Step 1: Let u= 3x2−5x+ 2.
Step 2: Differentiate uwith respect to xto get du
dx .
du
dx = 6x−5
Step 3: Apply the chain rule to differentiate the function f(x) =
ln(u).
df
dx =1
u
·
du
dx
Substitute u= 3x2−5x+ 2 and du
dx = 6x−5into the formula:
df
dx =1
3x2−5x+ 2
·(6x−5)
Therefore, the derivative of the function f(x) = ln(3x2−5x+ 2) with
respect to xis:
14
df
dx =6x−5
3x2−5x+ 2
This is the final answer.Sure! Here is the question along with its
step-by-step solution in LateX code:
Question 15: Differentiate the following function with respect to
x:
f(x) = ln(3x2
−5x+ 2)
Solution: To differentiate the given function f(x) = ln(3x2−5x+ 2)
with respect to x, we will use the chain rule of differentiation.
Step 1: Let u= 3x2−5x+ 2.
Step 2: Differentiate uwith respect to xto get du
dx .
du
dx = 6x−5
Step 3: Apply the chain rule to differentiate the function f(x) =
ln(u).
df
dx =1
u
·
du
dx
Substitute u= 3x2−5x+ 2 and du
dx = 6x−5into the formula:
df
dx =1
3x2−5x+ 2
·(6x−5)
Therefore, the derivative of the function f(x) = ln(3x2−5x+ 2) with
respect to xis:
df
dx =6x−5
3x2−5x+ 2
This is the final answer.
Question 16
y= 3 ln(x)−2 ln(x2)
Step-by-step solution: To differentiate the given function, we will
apply the properties of logarithms and the chain rule of differentia-
tion.
1. Given Function:
y= 3 ln(x)−2 ln(x2)
15
2. Apply the properties of logarithms:
y= ln(x3)−ln(x4)
y= ln(x3
x4)
y= ln(x−1)
3. Rewrite the function:
y=−ln(x)
4. Differentiate the function using the differentiation rule for ln(x):
dy
dx =−
1
x
5. Hence, the derivative of the given logarithmic function with
base eis: dy
dx =−
1
x
Therefore, the derivative of 3 ln(x)−2 ln(x2)with respect to xis
−
1
x.Question 16: Differentiate the following logarithmic function with
base e:
y= 3 ln(x)−2 ln(x2)
Step-by-step solution: To differentiate the given function, we will
apply the properties of logarithms and the chain rule of differentia-
tion.
1. Given Function:
y= 3 ln(x)−2 ln(x2)
2. Apply the properties of logarithms:
y= ln(x3)−ln(x4)
y= ln(x3
x4)
y= ln(x−1)
3. Rewrite the function:
y=−ln(x)
4. Differentiate the function using the differentiation rule for ln(x):
dy
dx =−
1
x
5. Hence, the derivative of the given logarithmic function with
base eis: dy
dx =−
1
x
Therefore, the derivative of 3 ln(x)−2 ln(x2)with respect to xis −
1
x.
16
Question 17
Differentiate the following function with respect to x:f(x) = 3 ln(4x2−
5).
Solution:
To differentiate the given function f(x) = 3 ln(4x2−5) with respect
to x, we will use the rule for differentiating logarithmic functions with
base e.
1. Let’s rewrite the function in a different form to make the differ-
entiation process easier: f(x) = 3 ln(4x2−5) = 3 ln(u), where u= 4x2−5.
2. Now, differentiate the function f(x) = 3 ln(u)using the chain
rule:
d
dx [3 ln(u)] = 3 ·
d
dx [ln(u)]
= 3 ·
1
u
·
du
dx
= 3 ·
1
4x2−5
·8x
=24x
4x2−5
3. Therefore, the derivative of the function f(x) = 3 ln(4x2−5) with
respect to xis:
d
dx 3 ln(4x2
−5)=24x
4x2−5
Question 17:
Differentiate the following function with respect to x:f(x) = 3 ln(4x2−
5).
Solution:
To differentiate the given function f(x) = 3 ln(4x2−5) with respect
to x, we will use the rule for differentiating logarithmic functions with
base e.
1. Let’s rewrite the function in a different form to make the differ-
entiation process easier: f(x) = 3 ln(4x2−5) = 3 ln(u), where u= 4x2−5.
2. Now, differentiate the function f(x) = 3 ln(u)using the chain
rule:
d
dx [3 ln(u)] = 3 ·
d
dx [ln(u)]
= 3 ·
1
u
·
du
dx
= 3 ·
1
4x2−5
·8x
=24x
4x2−5
17
3. Therefore, the derivative of the function f(x) = 3 ln(4x2−5) with
respect to xis:
d
dx 3 ln(4x2
−5)=24x
4x2−5
Question 18
Step-by-step solution: To differentiate the given function f(x) =
3 ln(x2+ 1), we will apply the rules of differentiation for logarithmic
functions with base e(natural logarithm).
1. Use the chain rule: differentiate the outer function (ln) first,
then multiply by the derivative of the inner function (x2+ 1).
2. The derivative of ln(u)with respect to xis 1
u
·du
dx .
3. Let u=x2+ 1. Then, du
dx = 2x.
4. Plug these values into the chain rule formula.
5. We get:
f′(x)=3·
1
x2+ 1
·2x
f′(x)=3·
2x
x2+ 1
f′(x) = 6x
x2+ 1
Therefore, the derivative of the function f(x) = 3 ln(x2+1) is 6x
x2+1 .Question
18: Differentiate the function f(x) = 3 ln(x2+ 1).
Step-by-step solution: To differentiate the given function f(x) =
3 ln(x2+ 1), we will apply the rules of differentiation for logarithmic
functions with base e(natural logarithm).
1. Use the chain rule: differentiate the outer function (ln) first,
then multiply by the derivative of the inner function (x2+ 1).
2. The derivative of ln(u)with respect to xis 1
u
·du
dx .
3. Let u=x2+ 1. Then, du
dx = 2x.
4. Plug these values into the chain rule formula.
5. We get:
f′(x)=3·
1
x2+ 1
·2x
f′(x)=3·
2x
x2+ 1
f′(x) = 6x
x2+ 1
Therefore, the derivative of the function f(x) = 3 ln(x2+ 1) is 6x
x2+1 .
18
Question 19
Differentiate the following function with respect to x, where f(x) =
3 ln(2x3+ 5x).
Solution:
Given function: f(x) = 3 ln(2x3+ 5x).
To differentiate a logarithmic function with base e, we use the
chain rule. The derivative of ln(u)with respect to xis 1
u
·du
dx .
Let u= 2x3+ 5x.
Now, we can differentiate f(x)using the chain rule:
df
dx = 3 ·
1
2x3+ 5x
·(6x2+ 5)
df
dx =3(6x2+ 5)
2x3+ 5x
Therefore, the derivative of f(x) = 3 ln(2x3+ 5x)with respect to x
is 18x2+15
2x3+5x.Question 19:
Differentiate the following function with respect to x, where f(x) =
3 ln(2x3+ 5x).
Solution:
Given function: f(x) = 3 ln(2x3+ 5x).
To differentiate a logarithmic function with base e, we use the
chain rule. The derivative of ln(u)with respect to xis 1
u
·du
dx .
Let u= 2x3+ 5x.
Now, we can differentiate f(x)using the chain rule:
df
dx = 3 ·
1
2x3+ 5x
·(6x2+ 5)
df
dx =3(6x2+ 5)
2x3+ 5x
Therefore, the derivative of f(x) = 3 ln(2x3+ 5x)with respect to x
is 18x2+15
2x3+5x.
Question 20
Solution: To differentiate the given function using logarithmic dif-
ferentiation, we follow these steps:
1. Let y= ln(3x2−2x).
2. Take the natural logarithm of both sides: ln(y) = ln(ln(3x2−2x)).
19
3. Differentiate both sides with respect to x:
d
dx [ln(y)] = d
dx [ln(3x2
−2x)]
1
y
dy
dx =1
3x2−2x
·(6x−2)
dy
dx =y·
6x−2
3x2−2x
= (ln(3x2
−2x)) ·
6x−2
3x2−2x.
4. Substitute back y= ln(3x2−2x)to get the final answer:
f′(x) = ln(3x2
−2x)·
6x−2
3x2−2x.
Question 20: Differentiate the following function with respect to
xusing logarithmic differentiation: f(x) = ln(3x2−2x).
Solution: To differentiate the given function using logarithmic dif-
ferentiation, we follow these steps:
1. Let y= ln(3x2−2x).
2. Take the natural logarithm of both sides: ln(y) = ln(ln(3x2−2x)).
3. Differentiate both sides with respect to x:
d
dx [ln(y)] = d
dx [ln(3x2
−2x)]
1
y
dy
dx =1
3x2−2x
·(6x−2)
dy
dx =y·
6x−2
3x2−2x
= (ln(3x2
−2x)) ·
6x−2
3x2−2x.
4. Substitute back y= ln(3x2−2x)to get the final answer:
f′(x) = ln(3x2
−2x)·
6x−2
3x2−2x.
20
To differentiate the function f(x) = ln(2x+ 1) with base e, we use
the formula for differentiating logarithmic functions:
d
dx ln(u) = 1
u
·
du
dx
In this case, our function is u= 2x+ 1, so we have:
d
dx ln(2x+ 1) = 1
2x+ 1
·
d
dx (2x+ 1)
Now, we find the derivative of 2x+ 1 with respect to x:
d
dx (2x+ 1) = 2
Plugging this back into our formula, we get:
d
dx ln(2x+ 1) = 1
2x+ 1
·2 = 2
2x+ 1
Therefore, the derivative of the function f(x) = ln(2x+ 1) with base
eis:
d
dx ln(2x+ 1) = 2
2x+ 1
Question 2
y= 5 ln(3x2
−4x)
Solution: To differentiate the given function y= 5 ln(3x2−4x)with
base e, we use the chain rule since the function is the natural loga-
rithm, which has a base of e.
Let’s denote u= 3x2−4x, so that y= 5 ln(u).
Now, we can differentiate using the chain rule, which states that
if y= ln(u), then dy
dx =1
u
du
dx .
1. Find dy
dx using the chain rule:
dy
dx = 5 ·
1
u
·
du
dx =5
3x2−4x
·(6x−4)
2. Simplify the expression:
dy
dx =5(6x−4)
3x2−4x=30x−20
3x2−4x
Therefore, the derivative of the function y= 5 ln(3x2−4x)with re-
spect to xis 30x−20
3x2−4x.Question 2: Differentiate the following logarithmic
function with base e:
2
y= 5 ln(3x2
−4x)
Solution: To differentiate the given function y= 5 ln(3x2−4x)with
base e, we use the chain rule since the function is the natural loga-
rithm, which has a base of e.
Let’s denote u= 3x2−4x, so that y= 5 ln(u).
Now, we can differentiate using the chain rule, which states that
if y= ln(u), then dy
dx =1
u
du
dx .
1. Find dy
dx using the chain rule:
dy
dx = 5 ·
1
u
·
du
dx =5
3x2−4x
·(6x−4)
2. Simplify the expression:
dy
dx =5(6x−4)
3x2−4x=30x−20
3x2−4x
Therefore, the derivative of the function y= 5 ln(3x2−4x)with
respect to xis 30x−20
3x2−4x.
Question 3
Step-by-step solution: To differentiate a logarithmic function with
base e, we use the formula d
dx (ln(u)) = 1
u
du
dx .
1. Differentiate the first term 3 ln(2x):
y1= 3 ln(2x)
dy1
dx = 3 ·
1
2x
·2
dy1
dx = 3 ·
1
x
2. Differentiate the second term 5 ln(x2):
y2= 5 ln(x2)
dy2
dx = 5 ·
1
x2
·2x
dy2
dx = 10 ·
1
x
3. Add the derivatives of the individual terms to find the overall
derivative of the function:
dy
dx =dy1
dx +dy2
dx
3
dy
dx = 3 ·
1
x+ 10 ·
1
x
dy
dx =3 + 10
x
dy
dx =13
x
Therefore, the derivative of the function y= 3 ln(2x) + 5 ln(x2)with
base eis 13
x.Question 3: Differentiate the following function with base
e:
y= 3 ln(2x) + 5 ln(x2)
Step-by-step solution: To differentiate a logarithmic function with
base e, we use the formula d
dx (ln(u)) = 1
u
du
dx .
1. Differentiate the first term 3 ln(2x):
y1= 3 ln(2x)
dy1
dx = 3 ·
1
2x
·2
dy1
dx = 3 ·
1
x
2. Differentiate the second term 5 ln(x2):
y2= 5 ln(x2)
dy2
dx = 5 ·
1
x2
·2x
dy2
dx = 10 ·
1
x
3. Add the derivatives of the individual terms to find the overall
derivative of the function:
dy
dx =dy1
dx +dy2
dx
dy
dx = 3 ·
1
x+ 10 ·
1
x
dy
dx =3 + 10
x
dy
dx =13
x
Therefore, the derivative of the function y= 3 ln(2x) + 5 ln(x2)with
base eis 13
x.
4
Question 4
Step-by-step solution: To differentiate the function f(x) = ln(2x3−
5x+ 1) with base e, we use the chain rule along with the derivative of
the natural logarithm function.
1. Let u= 2x3−5x+ 1. 2. Find the derivative of uwith respect to
x:du
dx = 6x2
−5
Now, differentiate the function with base eusing the chain rule:
d
dx [ln(u)] = 1
u
·
du
dx
d
dx [ln(2x3
−5x+ 1)] = 1
2x3−5x+ 1
·(6x2
−5)
Therefore, the derivative of f(x) = ln(2x3−5x+ 1) with base eis:
d
dx [ln(2x3
−5x+ 1)] = 6x2−5
2x3−5x+ 1
Question 4: Differentiate the following function with base e, where
f(x) = ln(2x3−5x+ 1).
Step-by-step solution: To differentiate the function f(x) = ln(2x3−
5x+ 1) with base e, we use the chain rule along with the derivative of
the natural logarithm function.
1. Let u= 2x3−5x+ 1. 2. Find the derivative of uwith respect to
x:du
dx = 6x2
−5
Now, differentiate the function with base eusing the chain rule:
d
dx [ln(u)] = 1
u
·
du
dx
d
dx [ln(2x3
−5x+ 1)] = 1
2x3−5x+ 1
·(6x2
−5)
Therefore, the derivative of f(x) = ln(2x3−5x+ 1) with base eis:
d
dx [ln(2x3
−5x+ 1)] = 6x2−5
2x3−5x+ 1
Question 5
Step-by-step solution: To differentiate the logarithmic function
with base e, we apply the chain rule.
Given function: y= ln(3x2+ 5x)
5
1. Apply the chain rule:
dy
dx =1
3x2+ 5x
·(6x+ 5)
dy
dx =6x+ 5
3x2+ 5x
Therefore, the derivative of y= ln(3x2+ 5x)with respect to xis
6x+5
3x2+5x.Question 5: Differentiate the following logarithmic function
with base e:y= ln(3x2+ 5x)
Step-by-step solution: To differentiate the logarithmic function
with base e, we apply the chain rule.
Given function: y= ln(3x2+ 5x)
1. Apply the chain rule:
dy
dx =1
3x2+ 5x
·(6x+ 5)
dy
dx =6x+ 5
3x2+ 5x
Therefore, the derivative of y= ln(3x2+ 5x)with respect to xis
6x+5
3x2+5x.
Question 6
Differentiate the following function with respect to x:
f(x) = 2 ln(e3x−5)
Solution:
To differentiate the function f(x) = 2 ln(e3x−5), we will use the
properties of logarithms and the chain rule.
1. Let’s first rewrite the function using the property ln(a) = ln(ea):
f(x) = 2 ln(e3x−5) = 2 ln(e3x−5)
2. Apply the chain rule to differentiate f(x) = 2 ln(e3x−5):
d
dx [2 ln(e3x−5)] = 2 ·
1
e3x−5
·
d
dx (e3x−5)
3. Find the derivative of e3x−5using the chain rule:
d
dx (e3x−5) = 3e3x
4. Substitute the derivative back into the expression:
2·
1
e3x−5
·3e3x
6
5. Simplify the expression:
6·
e3x
e3x−5
Therefore, the derivative of the function f(x) = 2 ln(e3x−5) is:
6e3x
e3x−5
Question 6:
Differentiate the following function with respect to x:
f(x) = 2 ln(e3x−5)
Solution:
To differentiate the function f(x) = 2 ln(e3x−5), we will use the
properties of logarithms and the chain rule.
1. Let’s first rewrite the function using the property ln(a) = ln(ea):
f(x) = 2 ln(e3x−5) = 2 ln(e3x−5)
2. Apply the chain rule to differentiate f(x) = 2 ln(e3x−5):
d
dx [2 ln(e3x−5)] = 2 ·
1
e3x−5
·
d
dx (e3x−5)
3. Find the derivative of e3x−5using the chain rule:
d
dx (e3x−5) = 3e3x
4. Substitute the derivative back into the expression:
2·
1
e3x−5
·3e3x
5. Simplify the expression:
6·
e3x
e3x−5
Therefore, the derivative of the function f(x) = 2 ln(e3x−5) is:
6e3x
e3x−5
7
Question 7
f(x) = 4 ln(3x)
Step-by-step solution: To differentiate the given function f(x) =
4 ln(3x)with base e, we will use the chain rule.
Step 1: Identify the function and its derivative. Let u= 3xand
y= ln(u).
Step 2: Find dy
du and du
dx . Since y= ln(u), we have dy
du =1
uand du
dx = 3.
Step 3: Apply the chain rule.
df
dx =df
du
·
du
dx = 4 ·
1
3x
·3=4·
1
x=4
x
Therefore, the derivative of the function f(x) = 4 ln(3x)with base e
is 4
x.Question 7: Differentiate the following logarithmic function with
base e:
f(x) = 4 ln(3x)
Step-by-step solution: To differentiate the given function f(x) =
4 ln(3x)with base e, we will use the chain rule.
Step 1: Identify the function and its derivative. Let u= 3xand
y= ln(u).
Step 2: Find dy
du and du
dx . Since y= ln(u), we have dy
du =1
uand du
dx = 3.
Step 3: Apply the chain rule.
df
dx =df
du
·
du
dx = 4 ·
1
3x
·3=4·
1
x=4
x
Therefore, the derivative of the function f(x) = 4 ln(3x)with base
eis 4
x.
Question 8
Solution: To differentiate the given function, we will apply the
product rule, which states that if f(x) = u(x)v(x), then f′(x) = u(x)v′(x)+
v(x)u′(x).
Given function: f(x)=5e3xln(2x)
Let u(x)=5e3xand v(x) = ln(2x).
Now, we differentiate u(x)and v(x)separately:
u′(x) = 5 ·3e3x= 15e3x
v′(x) = 1
2x
Now, we use the product rule to find the derivative of f(x):
f′(x) = u(x)v′(x) + v(x)u′(x)
f′(x) = (5e3x)1
2x+ ln(2x)(15e3x)
f′(x) = 5e3x
2x+ 15e3xln(2x)
8
Therefore, the derivative of the function f(x)=5e3xln(2x)with
respect to xis 5e3x
2x+15e3xln(2x).Question 8: Differentiate the function
f(x) = 5e3xln(2x)with respect to x.
Solution: To differentiate the given function, we will apply the
product rule, which states that if f(x) = u(x)v(x), then f′(x) = u(x)v′(x)+
v(x)u′(x).
Given function: f(x)=5e3xln(2x)
Let u(x)=5e3xand v(x) = ln(2x).
Now, we differentiate u(x)and v(x)separately:
u′(x) = 5 ·3e3x= 15e3x
v′(x) = 1
2x
Now, we use the product rule to find the derivative of f(x):
f′(x) = u(x)v′(x) + v(x)u′(x)
f′(x) = (5e3x)1
2x+ ln(2x)(15e3x)
f′(x) = 5e3x
2x+ 15e3xln(2x)
Therefore, the derivative of the function f(x)=5e3xln(2x)with
respect to xis 5e3x
2x+ 15e3xln(2x).
Question 9
Solution: To differentiate the given function f(x) = 3 ln(x2)with
respect to x, we will use the properties of logarithmic functions and
the chain rule.
1. Recall the derivative of the natural logarithm function: d
dx (ln(u)) =
1
u
·du
dx
2. Applying the chain rule, we have:
d
dx (3 ln(x2)) = 3 ·
1
x2
·
d
dx (x2)
3. Simplifying the expression, we get:
d
dx (3 ln(x2)) = 3 ·
1
x2
·2x
4. Further simplifying, we have:
d
dx (3 ln(x2)) = 6 ·
1
x
5. Therefore, the derivative of the function f(x) = 3 ln(x2)with
respect to xis:
d
dx (3 ln(x2)) = 6
x
Question 9: Differentiate the following function with respect to x:
f(x) = 3 ln(x2)
9
Solution: To differentiate the given function f(x) = 3 ln(x2)with
respect to x, we will use the properties of logarithmic functions and
the chain rule.
1. Recall the derivative of the natural logarithm function: d
dx (ln(u)) =
1
u
·du
dx
2. Applying the chain rule, we have:
d
dx (3 ln(x2)) = 3 ·
1
x2
·
d
dx (x2)
3. Simplifying the expression, we get:
d
dx (3 ln(x2)) = 3 ·
1
x2
·2x
4. Further simplifying, we have:
d
dx (3 ln(x2)) = 6 ·
1
x
5. Therefore, the derivative of the function f(x) = 3 ln(x2)with
respect to xis:
d
dx (3 ln(x2)) = 6
x
Question 10
Step-by-step solution: To differentiate the function f(x) = ln(2x2),
we use the fact that the derivative of ln(u)is 1
u
·u′, where uis a function
of x.
1. Apply the differentiation rule:
f′(x) = 1
2x2
·(2x)
f′(x) = 2x
2x2
f′(x) = 1
x
Therefore, the derivative of f(x) = ln(2x2)with respect to xis
f′(x) = 1
x.Question 10: Differentiate the following logarithmic function
with base e:
f(x) = ln(2x2)
Step-by-step solution: To differentiate the function f(x) = ln(2x2),
we use the fact that the derivative of ln(u)is 1
u
·u′, where uis a function
of x.
10
1. Apply the differentiation rule:
f′(x) = 1
2x2
·(2x)
f′(x) = 2x
2x2
f′(x) = 1
x
Therefore, the derivative of f(x) = ln(2x2)with respect to xis
f′(x) = 1
x.
Question 11
Step-by-Step Solution: Given function: f(x) = 3 ln(2x)
Applying the properties of logarithms: f(x) = 3 ln(2) + 3 ln(x)
Now, differentiate with respect to xusing the derivative of natural
logarithm: d
dx [3 ln(2) + 3 ln(x)]
Using the derivative of a constant times a function: d
dx [3 ln(2)] +
d
dx [3 ln(x)]
Since ln(2) is a constant, the derivative of a constant is 0: 0+3·
1
x
Simplify the expression: 3
x
Therefore, the derivative of the function f(x) = 3 ln(2x)is 3
x.Question
11: Find the derivative of the function f(x) = 3 ln(2x).
Step-by-Step Solution: Given function: f(x) = 3 ln(2x)
Applying the properties of logarithms: f(x) = 3 ln(2) + 3 ln(x)
Now, differentiate with respect to xusing the derivative of natural
logarithm: d
dx [3 ln(2) + 3 ln(x)]
Using the derivative of a constant times a function: d
dx [3 ln(2)] +
d
dx [3 ln(x)]
Since ln(2) is a constant, the derivative of a constant is 0: 0+3·
1
x
Simplify the expression: 3
x
Therefore, the derivative of the function f(x) = 3 ln(2x)is 3
x.
Question 12
Solution: To differentiate the function f(x) = ln(5x3), we use the
chain rule, which states that if we have a composition of functions
h(g(x)), then the derivative is given by (h′(g(x))) ·g′(x).
Given function: f(x) = ln(5x3).
Step 1: Identify the inner function g(x) = 5x3and the outer func-
tion h(x) = ln(x).
Step 2: Find the derivative of the inner function:
g′(x)=3·5x3−1= 15x2
11
Step 3: Find the derivative of the outer function:
h′(x) = 1
x
Step 4: Apply the chain rule:
f′(x) = h′(g(x)) ·g′(x) = 1
5x3
·15x2
Step 5: Simplify the expression:
f′(x) = 15x2
5x3=3
x
Therefore, the derivative of the function f(x) = ln(5x3)is f′(x) =
3
x.Question 12: Differentiate the function f(x) = ln(5x3).
Solution: To differentiate the function f(x) = ln(5x3), we use the
chain rule, which states that if we have a composition of functions
h(g(x)), then the derivative is given by (h′(g(x))) ·g′(x).
Given function: f(x) = ln(5x3).
Step 1: Identify the inner function g(x) = 5x3and the outer func-
tion h(x) = ln(x).
Step 2: Find the derivative of the inner function:
g′(x)=3·5x3−1= 15x2
Step 3: Find the derivative of the outer function:
h′(x) = 1
x
Step 4: Apply the chain rule:
f′(x) = h′(g(x)) ·g′(x) = 1
5x3
·15x2
Step 5: Simplify the expression:
f′(x) = 15x2
5x3=3
x
Therefore, the derivative of the function f(x) = ln(5x3)is f′(x) = 3
x.
Question 13
Step-by-step Solution: To differentiate the given function f(x) =
ln(3x2+4x+1) with respect to x, we use the chain rule for differentiation
and the derivative of the natural logarithm function.
12
1. Use the chain rule:
f′(x) = 1
3x2+ 4x+ 1
·
d
dx (3x2+ 4x+ 1)
2. Differentiate the inside function:
d
dx (3x2+ 4x+ 1) = 6x+ 4
3. Substitute the derivative of the inside function back into the
chain rule equation:
f′(x) = 1
3x2+ 4x+ 1
·(6x+ 4)
4. Simplify the expression:
f′(x) = 6x+ 4
3x2+ 4x+ 1
Therefore, the derivative of the function f(x) = ln(3x2+ 4x+ 1)
with respect to xis 6x+4
3x2+4x+1 .Question 13: Differentiate the following
function with respect to x:
f(x) = ln(3x2+ 4x+ 1)
Step-by-step Solution: To differentiate the given function f(x) =
ln(3x2+4x+1) with respect to x, we use the chain rule for differentiation
and the derivative of the natural logarithm function.
1. Use the chain rule:
f′(x) = 1
3x2+ 4x+ 1
·
d
dx (3x2+ 4x+ 1)
2. Differentiate the inside function:
d
dx (3x2+ 4x+ 1) = 6x+ 4
3. Substitute the derivative of the inside function back into the
chain rule equation:
f′(x) = 1
3x2+ 4x+ 1
·(6x+ 4)
4. Simplify the expression:
f′(x) = 6x+ 4
3x2+ 4x+ 1
Therefore, the derivative of the function f(x) = ln(3x2+ 4x+ 1) with
respect to xis 6x+4
3x2+4x+1 .
13
Question 14
Step-by-step solution: To differentiate the function f(x) = ln(3x2−
4x), we will use the chain rule: 1. Let u= 3x2−4x. 2. Apply the
chain rule: d
dx (ln(u)) = 1
u
·du
dx . 3. Find the derivative of u:du
dx = 6x−4.
4. Substitute the derivative of uback into the chain rule formula:
d
dx (ln(3x2−4x)) = 1
3x2−4x
·(6x−4). 5. Simplify the expression: f′(x) =
6x−4
3x2−4x.
Therefore, the derivative of the function f(x) = ln(3x2−4x)with
respect to xis f′(x) = 6x−4
3x2−4x.Question 14: Differentiate the following
logarithmic function with base e:f(x) = ln(3x2−4x)
Step-by-step solution: To differentiate the function f(x) = ln(3x2−
4x), we will use the chain rule: 1. Let u= 3x2−4x. 2. Apply the
chain rule: d
dx (ln(u)) = 1
u
·du
dx . 3. Find the derivative of u:du
dx = 6x−4.
4. Substitute the derivative of uback into the chain rule formula:
d
dx (ln(3x2−4x)) = 1
3x2−4x
·(6x−4). 5. Simplify the expression: f′(x) =
6x−4
3x2−4x.
Therefore, the derivative of the function f(x) = ln(3x2−4x)with
respect to xis f′(x) = 6x−4
3x2−4x.
Question 15
Question 15: Differentiate the following function with respect to
x:
f(x) = ln(3x2
−5x+ 2)
Solution: To differentiate the given function f(x) = ln(3x2−5x+ 2)
with respect to x, we will use the chain rule of differentiation.
Step 1: Let u= 3x2−5x+ 2.
Step 2: Differentiate uwith respect to xto get du
dx .
du
dx = 6x−5
Step 3: Apply the chain rule to differentiate the function f(x) =
ln(u).
df
dx =1
u
·
du
dx
Substitute u= 3x2−5x+ 2 and du
dx = 6x−5into the formula:
df
dx =1
3x2−5x+ 2
·(6x−5)
Therefore, the derivative of the function f(x) = ln(3x2−5x+ 2) with
respect to xis:
14
df
dx =6x−5
3x2−5x+ 2
This is the final answer.Sure! Here is the question along with its
step-by-step solution in LateX code:
Question 15: Differentiate the following function with respect to
x:
f(x) = ln(3x2
−5x+ 2)
Solution: To differentiate the given function f(x) = ln(3x2−5x+ 2)
with respect to x, we will use the chain rule of differentiation.
Step 1: Let u= 3x2−5x+ 2.
Step 2: Differentiate uwith respect to xto get du
dx .
du
dx = 6x−5
Step 3: Apply the chain rule to differentiate the function f(x) =
ln(u).
df
dx =1
u
·
du
dx
Substitute u= 3x2−5x+ 2 and du
dx = 6x−5into the formula:
df
dx =1
3x2−5x+ 2
·(6x−5)
Therefore, the derivative of the function f(x) = ln(3x2−5x+ 2) with
respect to xis:
df
dx =6x−5
3x2−5x+ 2
This is the final answer.
Question 16
y= 3 ln(x)−2 ln(x2)
Step-by-step solution: To differentiate the given function, we will
apply the properties of logarithms and the chain rule of differentia-
tion.
1. Given Function:
y= 3 ln(x)−2 ln(x2)
15
2. Apply the properties of logarithms:
y= ln(x3)−ln(x4)
y= ln(x3
x4)
y= ln(x−1)
3. Rewrite the function:
y=−ln(x)
4. Differentiate the function using the differentiation rule for ln(x):
dy
dx =−
1
x
5. Hence, the derivative of the given logarithmic function with
base eis: dy
dx =−
1
x
Therefore, the derivative of 3 ln(x)−2 ln(x2)with respect to xis
−
1
x.Question 16: Differentiate the following logarithmic function with
base e:
y= 3 ln(x)−2 ln(x2)
Step-by-step solution: To differentiate the given function, we will
apply the properties of logarithms and the chain rule of differentia-
tion.
1. Given Function:
y= 3 ln(x)−2 ln(x2)
2. Apply the properties of logarithms:
y= ln(x3)−ln(x4)
y= ln(x3
x4)
y= ln(x−1)
3. Rewrite the function:
y=−ln(x)
4. Differentiate the function using the differentiation rule for ln(x):
dy
dx =−
1
x
5. Hence, the derivative of the given logarithmic function with
base eis: dy
dx =−
1
x
Therefore, the derivative of 3 ln(x)−2 ln(x2)with respect to xis −
1
x.
16
Question 17
Differentiate the following function with respect to x:f(x) = 3 ln(4x2−
5).
Solution:
To differentiate the given function f(x) = 3 ln(4x2−5) with respect
to x, we will use the rule for differentiating logarithmic functions with
base e.
1. Let’s rewrite the function in a different form to make the differ-
entiation process easier: f(x) = 3 ln(4x2−5) = 3 ln(u), where u= 4x2−5.
2. Now, differentiate the function f(x) = 3 ln(u)using the chain
rule:
d
dx [3 ln(u)] = 3 ·
d
dx [ln(u)]
= 3 ·
1
u
·
du
dx
= 3 ·
1
4x2−5
·8x
=24x
4x2−5
3. Therefore, the derivative of the function f(x) = 3 ln(4x2−5) with
respect to xis:
d
dx 3 ln(4x2
−5)=24x
4x2−5
Question 17:
Differentiate the following function with respect to x:f(x) = 3 ln(4x2−
5).
Solution:
To differentiate the given function f(x) = 3 ln(4x2−5) with respect
to x, we will use the rule for differentiating logarithmic functions with
base e.
1. Let’s rewrite the function in a different form to make the differ-
entiation process easier: f(x) = 3 ln(4x2−5) = 3 ln(u), where u= 4x2−5.
2. Now, differentiate the function f(x) = 3 ln(u)using the chain
rule:
d
dx [3 ln(u)] = 3 ·
d
dx [ln(u)]
= 3 ·
1
u
·
du
dx
= 3 ·
1
4x2−5
·8x
=24x
4x2−5
17
3. Therefore, the derivative of the function f(x) = 3 ln(4x2−5) with
respect to xis:
d
dx 3 ln(4x2
−5)=24x
4x2−5
Question 18
Step-by-step solution: To differentiate the given function f(x) =
3 ln(x2+ 1), we will apply the rules of differentiation for logarithmic
functions with base e(natural logarithm).
1. Use the chain rule: differentiate the outer function (ln) first,
then multiply by the derivative of the inner function (x2+ 1).
2. The derivative of ln(u)with respect to xis 1
u
·du
dx .
3. Let u=x2+ 1. Then, du
dx = 2x.
4. Plug these values into the chain rule formula.
5. We get:
f′(x)=3·
1
x2+ 1
·2x
f′(x)=3·
2x
x2+ 1
f′(x) = 6x
x2+ 1
Therefore, the derivative of the function f(x) = 3 ln(x2+1) is 6x
x2+1 .Question
18: Differentiate the function f(x) = 3 ln(x2+ 1).
Step-by-step solution: To differentiate the given function f(x) =
3 ln(x2+ 1), we will apply the rules of differentiation for logarithmic
functions with base e(natural logarithm).
1. Use the chain rule: differentiate the outer function (ln) first,
then multiply by the derivative of the inner function (x2+ 1).
2. The derivative of ln(u)with respect to xis 1
u
·du
dx .
3. Let u=x2+ 1. Then, du
dx = 2x.
4. Plug these values into the chain rule formula.
5. We get:
f′(x)=3·
1
x2+ 1
·2x
f′(x)=3·
2x
x2+ 1
f′(x) = 6x
x2+ 1
Therefore, the derivative of the function f(x) = 3 ln(x2+ 1) is 6x
x2+1 .
18
Question 19
Differentiate the following function with respect to x, where f(x) =
3 ln(2x3+ 5x).
Solution:
Given function: f(x) = 3 ln(2x3+ 5x).
To differentiate a logarithmic function with base e, we use the
chain rule. The derivative of ln(u)with respect to xis 1
u
·du
dx .
Let u= 2x3+ 5x.
Now, we can differentiate f(x)using the chain rule:
df
dx = 3 ·
1
2x3+ 5x
·(6x2+ 5)
df
dx =3(6x2+ 5)
2x3+ 5x
Therefore, the derivative of f(x) = 3 ln(2x3+ 5x)with respect to x
is 18x2+15
2x3+5x.Question 19:
Differentiate the following function with respect to x, where f(x) =
3 ln(2x3+ 5x).
Solution:
Given function: f(x) = 3 ln(2x3+ 5x).
To differentiate a logarithmic function with base e, we use the
chain rule. The derivative of ln(u)with respect to xis 1
u
·du
dx .
Let u= 2x3+ 5x.
Now, we can differentiate f(x)using the chain rule:
df
dx = 3 ·
1
2x3+ 5x
·(6x2+ 5)
df
dx =3(6x2+ 5)
2x3+ 5x
Therefore, the derivative of f(x) = 3 ln(2x3+ 5x)with respect to x
is 18x2+15
2x3+5x.
Question 20
Solution: To differentiate the given function using logarithmic dif-
ferentiation, we follow these steps:
1. Let y= ln(3x2−2x).
2. Take the natural logarithm of both sides: ln(y) = ln(ln(3x2−2x)).
19
3. Differentiate both sides with respect to x:
d
dx [ln(y)] = d
dx [ln(3x2
−2x)]
1
y
dy
dx =1
3x2−2x
·(6x−2)
dy
dx =y·
6x−2
3x2−2x
= (ln(3x2
−2x)) ·
6x−2
3x2−2x.
4. Substitute back y= ln(3x2−2x)to get the final answer:
f′(x) = ln(3x2
−2x)·
6x−2
3x2−2x.
Question 20: Differentiate the following function with respect to
xusing logarithmic differentiation: f(x) = ln(3x2−2x).
Solution: To differentiate the given function using logarithmic dif-
ferentiation, we follow these steps:
1. Let y= ln(3x2−2x).
2. Take the natural logarithm of both sides: ln(y) = ln(ln(3x2−2x)).
3. Differentiate both sides with respect to x:
d
dx [ln(y)] = d
dx [ln(3x2
−2x)]
1
y
dy
dx =1
3x2−2x
·(6x−2)
dy
dx =y·
6x−2
3x2−2x
= (ln(3x2
−2x)) ·
6x−2
3x2−2x.
4. Substitute back y= ln(3x2−2x)to get the final answer:
f′(x) = ln(3x2
−2x)·
6x−2
3x2−2x.
20
