MATH 332 - Differentiation - Chain Rule
Question Bank
Question 1
Problem:
Differentiate the function f(x) = sin(3x2+ 2x) using the chain rule.
—
Solution:
Step 1: Identify the inner and outer functions. In the given function f(x) =
sin(3x2+ 2x), the outer function, g(u), is sin(u), and the inner function, u(x),
is 3x2+ 2x.
Step 2: Differentiate the outer function with respect to the inner function.
The derivative of the outer function sin(u) with respect to uis cos(u). Thus,
g′(u) = cos(u).
Step 3: Differentiate the inner function with respect to x. The inner function
u(x)=3x2+ 2xcan be differentiated as follows:
u′(x) = (3x2+ 2x)′= 6x+ 2.
Step 4: Apply the chain rule. The derivative of the composite function f(x)
using the chain rule is given by:
f′(x) = g′(u(x)) ×u′(x) = cos(3x2+ 2x)×(6x+ 2).
Step 5: Simplify the expression (if possible).
f′(x) = (6x+ 2) cos(3x2+ 2x)
This is the derivative of the function f(x) = sin(3x2+ 2x) using the chain
rule. Question 1: Differentiation Using the Chain Rule
Problem:
Differentiate the function f(x) = sin(3x2+ 2x)using the chain rule.
—
Solution:
Step 1: Identify the inner and outer functions. In the given func-
tion f(x) = sin(3x2+ 2x), the outer function, g(u), is sin(u), and the
inner function, u(x), is 3x2+ 2x.
1
Step 2: Differentiate the outer function with respect to the inner
function. The derivative of the outer function sin(u)with respect to
uis cos(u). Thus, g′(u) = cos(u).
Step 3: Differentiate the inner function with respect to x. The
inner function u(x)=3x2+ 2xcan be differentiated as follows:
u′(x) = (3x2+ 2x)′= 6x+ 2.
Step 4: Apply the chain rule. The derivative of the composite
function f(x)using the chain rule is given by:
f′(x) = g′(u(x)) ×u′(x) = cos(3x2+ 2x)×(6x+ 2).
Step 5: Simplify the expression (if possible).
f′(x) = (6x+ 2) cos(3x2+ 2x)
This is the derivative of the function f(x) = sin(3x2+ 2x)using the
chain rule.
Question 2
Given the function f(x) = √3x2+ 2x+ 1, find f′(x), the derivative
of the function with respect to x.
—
Solution:
Step 1: Identify the Outer and Inner Functions - The outer func-
tion g(u)is √u- The inner function u(x)is 3x2+ 2x+ 1
Step 2: Differentiate the Outer Function - The derivative of g(u) =
√uwith respect to uis g′(u) = 1
2√u
Step 3: Differentiate the Inner Function - The derivative of u(x) =
3x2+ 2x+ 1 with respect to xis u′(x)=6x+ 2
Step 4: Apply the Chain Rule - The chain rule states that f′(x) =
g′(u)·u′(x). Plug in the derivatives from Step 2 and Step 3:
f′(x) = 1
2√3x2+ 2x+ 1 ·(6x+ 2)
Step 5: Simplify the Expression - Combine and simplify the ex-
pression:
f′(x) = 6x+ 2
2√3x2+ 2x+ 1
- Further simplification gives:
f′(x) = 3x+ 1
√3x2+ 2x+ 1
2
Final Answer:
f′(Quick(x) = 3x+ 1
√3x2+ 2x+ 1
Question 2: Differentiate the function using the Chain Rule
Given the function f(x) = √3x2+ 2x+ 1, find f′(x), the derivative
of the function with respect to x.
—
Solution:
Step 1: Identify the Outer and Inner Functions - The outer func-
tion g(u)is √u- The inner function u(x)is 3x2+ 2x+ 1
Step 2: Differentiate the Outer Function - The derivative of g(u) =
√uwith respect to uis g′(u) = 1
2√u
Step 3: Differentiate the Inner Function - The derivative of u(x) =
3x2+ 2x+ 1 with respect to xis u′(x)=6x+ 2
Step 4: Apply the Chain Rule - The chain rule states that f′(x) =
g′(u)·u′(x). Plug in the derivatives from Step 2 and Step 3:
f′(x) = 1
2√3x2+ 2x+ 1 ·(6x+ 2)
Step 5: Simplify the Expression - Combine and simplify the ex-
pression:
f′(x) = 6x+ 2
2√3x2+ 2x+ 1
- Further simplification gives:
f′(x) = 3x+ 1
√3x2+ 2x+ 1
Final Answer:
f′(Quick(x) = 3x+ 1
√3x2+ 2x+ 1
Question 3
Find the derivative y′of the function y= sin(e2x).
Solution:
Step 1: Recognize the outer function and the inner function. -
Let u=e2x, which is the inner function. - The outer function then
becomes y= sin(u).
Step 2: Differentiate the outer function ywith respect to u. -
Differentiate y= sin(u)with respect to uto get dy
du = cos(u).
Step 3: Differentiate the inner function uwith respect to x. -
Differentiate u=e2xwith respect to xto get du
dx = 2e2x.
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Step 4: Apply the chain rule. - Multiply the derivatives from Step
2 and Step 3 to get the derivative of ywith respect to x.
y′=dy
dx =dy
du ·du
dx = cos(u)·2e2x
Step 5: Substitute back the expression for u. - Substitute u=e2x
back into the derivative:
y′= 2e2xcos(e2x)
Final Answer: The derivative of y= sin(e2x)is y′= 2e2xcos(e2x).
Question:
Find the derivative y′of the function y= sin(e2x).
Solution:
Step 1: Recognize the outer function and the inner function. -
Let u=e2x, which is the inner function. - The outer function then
becomes y= sin(u).
Step 2: Differentiate the outer function ywith respect to u. -
Differentiate y= sin(u)with respect to uto get dy
du = cos(u).
Step 3: Differentiate the inner function uwith respect to x. -
Differentiate u=e2xwith respect to xto get du
dx = 2e2x.
Step 4: Apply the chain rule. - Multiply the derivatives from Step
2 and Step 3 to get the derivative of ywith respect to x.
y′=dy
dx =dy
du ·du
dx = cos(u)·2e2x
Step 5: Substitute back the expression for u. - Substitute u=e2x
back into the derivative:
y′= 2e2xcos(e2x)
Final Answer: The derivative of y= sin(e2x)is y′= 2e2xcos(e2x).
Question 4
Problem: Differentiate the function f(x) = sin(5x2+ 3x)using the
chain rule.
Solution:
Step 1: Identify the outer and inner functions. - Outer function,
g(u), where u= 5x2+ 3xis sin(u). - Inner function, u(x), is 5x2+ 3x.
Step 2: Differentiate the outer function with respect to the inner
function. - Differentiate g(u) = sin(u)with respect to u:
g′(u) = cos(u)
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Step 3: Differentiate the inner function with respect to x. - Dif-
ferentiate u(x)=5x2+ 3xwith respect to x:
u′(x) = 10x+ 3
Step 4: Apply the chain rule. - Multiply the derivatives obtained
in steps 2 and 3:
f′(x) = g′(u)·u′(x) = cos(5x2+ 3x)·(10x+ 3)
Step 5: Combine and express the final result. - Thus, the deriva-
tive of f(x)is:
f′(x) = cos(5x2+ 3x)·(10x+ 3)
This completes the differentiation process using the chain rule for
the given function. Question 4: Differentiation - Chain Rule
Problem: Differentiate the function f(x) = sin(5x2+ 3x)using the
chain rule.
Solution:
Step 1: Identify the outer and inner functions. - Outer function,
g(u), where u= 5x2+ 3xis sin(u). - Inner function, u(x), is 5x2+ 3x.
Step 2: Differentiate the outer function with respect to the inner
function. - Differentiate g(u) = sin(u)with respect to u:
g′(u) = cos(u)
Step 3: Differentiate the inner function with respect to x. - Dif-
ferentiate u(x) = 5x2+ 3xwith respect to x:
u′(x) = 10x+ 3
Step 4: Apply the chain rule. - Multiply the derivatives obtained
in steps 2 and 3:
f′(x) = g′(u)·u′(x) = cos(5x2+ 3x)·(10x+ 3)
Step 5: Combine and express the final result. - Thus, the deriva-
tive of f(x)is:
f′(x) = cos(5x2+ 3x)·(10x+ 3)
This completes the differentiation process using the chain rule for
the given function.
Question 5
Step-by-Step Solution:
Step 1: Recognize that the function f(x) = sin2(3x+ 5) can be
written as f(x) = [sin(3x+ 5)]2. Looking at f(x), identify the outer
function and the inner function needed for the chain rule.
5
- Outer function: u2where u= sin(3x+ 5). - Inner function: u=
sin(3x+ 5).
Step 2: Differentiate the outer function with respect to u:
d
du (u2)=2u.
Step 3: Differentiate the inner function with respect to x:
d
dx (sin(3x+ 5)) = cos(3x+ 5) ·d
dx (3x+ 5) = 3 cos(3x+ 5).
Step 4: Apply the chain rule. Multiply the derivative of the outer
function by the derivative of the inner function:
df
dx = 2(sin(3x+ 5)) ·3 cos(3x+ 5) = 6 sin(3x+ 5) cos(3x+ 5).
Step 5: Optionally, use the double angle identity: sin(2θ) = 2 sin(θ) cos(θ),
to simplify the expression wherein θ= 3x+ 5:
6 sin(3x+ 5) cos(3x+ 5) = 3 sin(2(3x+ 5)) = 3 sin(6x+ 10).
Final Answer:
df
dx = 3 sin(6x+ 10).
Question 5: Differentiate f(x) = sin2(3x+ 5).
Step-by-Step Solution:
Step 1: Recognize that the function f(x) = sin2(3x+ 5) can be
written as f(x) = [sin(3x+ 5)]2. Looking at f(x), identify the outer
function and the inner function needed for the chain rule.
- Outer function: u2where u= sin(3x+ 5). - Inner function: u=
sin(3x+ 5).
Step 2: Differentiate the outer function with respect to u:
d
du (u2)=2u.
Step 3: Differentiate the inner function with respect to x:
d
dx (sin(3x+ 5)) = cos(3x+ 5) ·d
dx (3x+ 5) = 3 cos(3x+ 5).
Step 4: Apply the chain rule. Multiply the derivative of the outer
function by the derivative of the inner function:
df
dx = 2(sin(3x+ 5)) ·3 cos(3x+ 5) = 6 sin(3x+ 5) cos(3x+ 5).
6
Step 5: Optionally, use the double angle identity: sin(2θ) = 2 sin(θ) cos(θ),
to simplify the expression wherein θ= 3x+ 5:
6 sin(3x+ 5) cos(3x+ 5) = 3 sin(2(3x+ 5)) = 3 sin(6x+ 10).
Final Answer:
df
dx = 3 sin(6x+ 10).
Question 6
Problem Statement: Find the derivative of the function f(x) =
sin(3x2+ 2x)using the chain rule.
—
Solution:
Step 1: Identify the outer and inner functions. - The outer func-
tion is g(u) = sin(u). - The inner function is h(x) = 3x2+ 2x.
Step 2: Differentiate the outer function with respect to the inner
function. - The derivative of the outer function g(u)with respect to
uis g′(u) = cos(u).
Step 3: Differentiate the inner function. - The derivative of h(x) =
3x2+ 2xwith respect to xis h′(x) = 6x+ 2 (using basic differentiation
rules).
Step 4: Apply the chain rule. - The chain rule states that (g◦
h)′(x) = g′(h(x)) ×h′(x). - Plugging in the derivatives you found, you
get:
f′(x) = cos(3x2+ 2x)·(6x+ 2)
Step 5: Simplify the expression. - The simplified form of the
derivative is:
f′(x) = (6x+ 2) cos(3x2+ 2x)
Final Answer: The derivative of the function f(x) = sin(3x2+ 2x)
is f′(x) = (6x+ 2) cos(3x2+ 2x). Question 6: Differentiation Using the
Chain Rule
Problem Statement: Find the derivative of the function f(x) =
sin(3x2+ 2x)using the chain rule.
—
Solution:
Step 1: Identify the outer and inner functions. - The outer func-
tion is g(u) = sin(u). - The inner function is h(x)=3x2+ 2x.
Step 2: Differentiate the outer function with respect to the inner
function. - The derivative of the outer function g(u)with respect to
uis g′(u) = cos(u).
Step 3: Differentiate the inner function. - The derivative of h(x) =
3x2+ 2xwith respect to xis h′(x) = 6x+ 2 (using basic differentiation
rules).
7
Step 4: Apply the chain rule. - The chain rule states that (g◦
h)′(x) = g′(h(x)) ×h′(x). - Plugging in the derivatives you found, you
get:
f′(x) = cos(3x2+ 2x)·(6x+ 2)
Step 5: Simplify the expression. - The simplified form of the
derivative is:
f′(x) = (6x+ 2) cos(3x2+ 2x)
Final Answer: The derivative of the function f(x) = sin(3x2+ 2x)is
f′(x) = (6x+ 2) cos(3x2+ 2x).
Question 7
Given the function f(x) = sin(5x2+ 3x−2), use the chain rule to
find the derivative f′(x).
Solution:
Step 1: Identify the Outer and Inner Functions - Outer function:
g(u) = sin(u), where uis the function inside the parentheses. - Inner
function: h(x) = 5x2+ 3x−2, which is the expression inside the sine
function.
Step 2: Differentiate the Outer Function - Differentiate g(u)with
respect to uto get g′(u) = cos(u).
Step 3: Differentiate the Inner Function - Differentiate h(x) =
5x2+ 3x−2with respect to x.
h′(x) = (5 ·2x)+3−0 = 10x+ 3
Step 4: Apply the Chain Rule - The chain rule states that f′(x) =
g′(h(x))·h′(x). - Substitute g′(h(x)) = cos(5x2+3x−2) and h′(x) = 10x+3.
Step 5: Write the Final Result - Combine the derivatives:
f′(x) = cos(5x2+ 3x−2) ·(10x+ 3)
This is the derivative of f(x)using the chain rule.
Additional Instruction Question:
Verify f′(x)by evaluating f′(2) using your result.
Solution:
Step 1: Substitute x= 2 into f′(x)- Replace xwith 2 in the deriva-
tive:
f′(2) = cos(5(2)2+ 3(2) −2) ·(10(2) + 3)
Step 2: Simplify the Calculation - Calculate inside the cosine and
then the multiplication:
f′(2) = cos(5 ·4+6−2) ·(20 + 3) = cos(20 + 4) ·23 = cos(24) ·23
8
Step 3: Numeric Evaluation (if a calculator is allowed) - Evaluate
cos(24) using a calculator and multiply by 23 to get the numerical
value.
f′(2) cos(24◦)·23
(Here, you might need a calculator to find the precise value, or you
can keep it in this symbolic form if the problem does not require a
numerical answer.)
These steps show how to differentiate a function using the chain
rule and to verify your result through a specific substitution. Ques-
tion 7: Differentiation Using the Chain Rule
Given the function f(x) = sin(5x2+ 3x−2), use the chain rule to
find the derivative f′(x).
Solution:
Step 1: Identify the Outer and Inner Functions - Outer function:
g(u) = sin(u), where uis the function inside the parentheses. - Inner
function: h(x) = 5x2+ 3x−2, which is the expression inside the sine
function.
Step 2: Differentiate the Outer Function - Differentiate g(u)with
respect to uto get g′(u) = cos(u).
Step 3: Differentiate the Inner Function - Differentiate h(x) =
5x2+ 3x−2with respect to x.
h′(x) = (5 ·2x)+3−0 = 10x+ 3
Step 4: Apply the Chain Rule - The chain rule states that f′(x) =
g′(h(x))·h′(x). - Substitute g′(h(x)) = cos(5x2+3x−2) and h′(x) = 10x+3.
Step 5: Write the Final Result - Combine the derivatives:
f′(x) = cos(5x2+ 3x−2) ·(10x+ 3)
This is the derivative of f(x)using the chain rule.
Additional Instruction Question:
Verify f′(x)by evaluating f′(2) using your result.
Solution:
Step 1: Substitute x= 2 into f′(x)- Replace xwith 2 in the deriva-
tive:
f′(2) = cos(5(2)2+ 3(2) −2) ·(10(2) + 3)
Step 2: Simplify the Calculation - Calculate inside the cosine and
then the multiplication:
f′(2) = cos(5 ·4+6−2) ·(20 + 3) = cos(20 + 4) ·23 = cos(24) ·23
Step 3: Numeric Evaluation (if a calculator is allowed) - Evaluate
cos(24) using a calculator and multiply by 23 to get the numerical
value.
f′(2) cos(24◦)·23
9
(Here, you might need a calculator to find the precise value, or you
can keep it in this symbolic form if the problem does not require a
numerical answer.)
These steps show how to differentiate a function using the chain
rule and to verify your result through a specific substitution.
Question 8
Step-by-Step Solution:
Step 1: Identify the outer and inner functions. - The outer func-
tion is g(u) = eu. - The inner function is u(x)=3x2+ 2x.
Step 2: Differentiate the outer function with respect to the inner
function. - The derivative of g(u) = euwith respect to uis g′(u) = eu.
Step 3: Differentiate the inner function with respect to x. - The
derivative of u(x)=3x2+ 2xwith respect to xis u′(x)=6x+ 2.
Step 4: Apply the chain rule. - The chain rule states that f′(x) =
g′(u(x)) ·u′(x).
Step 5: Substitute back the derivatives from steps 2 and 3. -
f′(x) = eu(x)·(6x+ 2).
Step 6: Substitute u(x)=3x2+ 2xback in. - f′(x) = e3x2+2x·(6x+ 2).
Answer: The derivative of the function f(x) = e3x2+2xis f′(x) =
(6x+ 2)e3x2+2x. Question: Determine the derivative of the function
f(x) = e3x2+2xusing the chain rule.
Step-by-Step Solution:
Step 1: Identify the outer and inner functions. - The outer func-
tion is g(u) = eu. - The inner function is u(x)=3x2+ 2x.
Step 2: Differentiate the outer function with respect to the inner
function. - The derivative of g(u) = euwith respect to uis g′(u) = eu.
Step 3: Differentiate the inner function with respect to x. - The
derivative of u(x)=3x2+ 2xwith respect to xis u′(x)=6x+ 2.
Step 4: Apply the chain rule. - The chain rule states that f′(x) =
g′(u(x)) ·u′(x).
Step 5: Substitute back the derivatives from steps 2 and 3. -
f′(x) = eu(x)·(6x+ 2).
Step 6: Substitute u(x)=3x2+ 2xback in. - f′(x) = e3x2+2x·(6x+ 2).
Answer: The derivative of the function f(x) = e3x2+2xis f′(x) =
(6x+ 2)e3x2+2x.
Question 9
Given the function f(x) = √5x2+ 4x−1, find the derivative f′(x).
Solution:
To find the derivative of the function f(x) = √5x2+ 4x−1, we will
use the chain rule. The chain rule states that the derivative of a
10
composite function is the derivative of the outer function evaluated at
the inner function, multiplied by the derivative of the inner function.
Step 1: Identify the inner and outer functions. - Inner function
u(x)=5x2+ 4x−1- Outer function g(u) = √u
Step 2: Differentiate the outer function with respect to the inner
function (u).
g(u) = √u
g′(u) = 1
2√u
Step 3: Differentiate the inner function with respect to x.
u(x) = 5x2+ 4x−1
u′(x) = 10x+ 4
Step 4: Apply the chain rule.
f′(x) = g′(u)·u′(x)
Substitute for g′(u)and u′(x):
f′(x) = 1
2√5x2+ 4x−1·(10x+ 4)
Step 5: Simplify the expression.
f′(x) = 10x+ 4
2√5x2+ 4x−1
Final answer:
f′(x) = 5x+ 2
√5x2+ 4x−1
This is the derivative of the function f(x) = √5x2+ 4x−1using the
chain rule. Question 9 - Differentiation Using the Chain Rule
Given the function f(x) = √5x2+ 4x−1, find the derivative f′(x).
Solution:
To find the derivative of the function f(x) = √5x2+ 4x−1, we will
use the chain rule. The chain rule states that the derivative of a
composite function is the derivative of the outer function evaluated at
the inner function, multiplied by the derivative of the inner function.
Step 1: Identify the inner and outer functions. - Inner function
u(x) = 5x2+ 4x−1- Outer function g(u) = √u
Step 2: Differentiate the outer function with respect to the inner
function (u).
g(u) = √u
g′(u) = 1
2√u
11
Step 3: Differentiate the inner function with respect to x.
u(x) = 5x2+ 4x−1
u′(x) = 10x+ 4
Step 4: Apply the chain rule.
f′(x) = g′(u)·u′(x)
Substitute for g′(u)and u′(x):
f′(x) = 1
2√5x2+ 4x−1·(10x+ 4)
Step 5: Simplify the expression.
f′(x) = 10x+ 4
2√5x2+ 4x−1
Final answer:
f′(x) = 5x+ 2
√5x2+ 4x−1
This is the derivative of the function f(x) = √5x2+ 4x−1using the
chain rule.
Question 10
Find the derivative of the function y=√3x4−5x2+ 2 using the
chain rule.
Solution
Step 1: Identify the outer and inner functions. - The outer func-
tion f(u)is √u. - The inner function g(x)is 3x4−5x2+ 2.
Step 2: Differentiate the outer function with respect to the inner
function. The derivative of f(u) = √uis f′(u) = 1
2√u.
Step 3: Differentiate the inner function with respect to x. The
derivative of g(x)=3x4−5x2+ 2 is:
g′(x) = 12x3−10x
Step 4: Apply the chain rule. The chain rule states that the
derivative of f(g(x)) is f′(g(x)) ·g′(x). Thus, applying the chain rule
gives:
y′=1
2√3x4−5x2+ 2 ·(12x3−10x)
Step 5: Simplify (if possible). To simplify, we can factor out the
common terms in the derivative of the inner function:
y′=1
2√3x4−5x2+ 2 ·2x(6x2−5)
12
y′=x(6x2−5)
√3x4−5x2+ 2
This is the derivative of the function using the chain rule. Question
10: Differentiation - Chain Rule
Find the derivative of the function y=√3x4−5x2+ 2 using the
chain rule.
Solution
Step 1: Identify the outer and inner functions. - The outer func-
tion f(u)is √u. - The inner function g(x)is 3x4−5x2+ 2.
Step 2: Differentiate the outer function with respect to the inner
function. The derivative of f(u) = √uis f′(u) = 1
2√u.
Step 3: Differentiate the inner function with respect to x. The
derivative of g(x)=3x4−5x2+ 2 is:
g′(x) = 12x3−10x
Step 4: Apply the chain rule. The chain rule states that the
derivative of f(g(x)) is f′(g(x)) ·g′(x). Thus, applying the chain rule
gives:
y′=1
2√3x4−5x2+ 2 ·(12x3−10x)
Step 5: Simplify (if possible). To simplify, we can factor out the
common terms in the derivative of the inner function:
y′=1
2√3x4−5x2+ 2 ·2x(6x2−5)
y′=x(6x2−5)
√3x4−5x2+ 2
This is the derivative of the function using the chain rule.
Question 11
Question: Find the derivative of the function f(x) = cos(3x2−2)
using the Chain Rule.
Solution:
Step 1: Identify the outer and inner functions.
The given function is f(x) = cos(3x2−2). Here, the outer function
is cos(u)where u= 3x2−2is the inner function.
Step 2: Differentiate the outer function with respect to the inner
function.
Let u= 3x2−2. The derivative of cos(u)with respect to uis −sin(u).
Step 3: Differentiate the inner function with respect to x.
The derivative of u= 3x2−2with respect to xis du
dx = 6x.
Step 4: Apply the Chain Rule.
13
The Chain Rule states that if a function yis composed of two func-
tions uand g(x)such that y=f(u)and u=g(x), then the derivative
of ywith respect to xis:
dy
dx =df
du ·du
dx
Plugging in the derivatives:
df
dx =−sin(u)·6x
Step 5: Substitute back the original function of u.
Remember, u= 3x2−2. So, substituting back:
df
dx =−sin(3x2−2) ·6x
Final Answer:
df
dx =−6xsin(3x2−2)
This is the derivative of the function f(x) = cos(3x2−2) using the
Chain Rule. Question 11: Differentiation - Chain Rule at Liberty
University
Question: Find the derivative of the function f(x) = cos(3x2−2)
using the Chain Rule.
Solution:
Step 1: Identify the outer and inner functions.
The given function is f(x) = cos(3x2−2). Here, the outer function
is cos(u)where u= 3x2−2is the inner function.
Step 2: Differentiate the outer function with respect to the inner
function.
Let u= 3x2−2. The derivative of cos(u)with respect to uis −sin(u).
Step 3: Differentiate the inner function with respect to x.
The derivative of u= 3x2−2with respect to xis du
dx = 6x.
Step 4: Apply the Chain Rule.
The Chain Rule states that if a function yis composed of two func-
tions uand g(x)such that y=f(u)and u=g(x), then the derivative
of ywith respect to xis:
dy
dx =df
du ·du
dx
Plugging in the derivatives:
df
dx =−sin(u)·6x
Step 5: Substitute back the original function of u.
Remember, u= 3x2−2. So, substituting back:
14
df
dx =−sin(3x2−2) ·6x
Final Answer:
df
dx =−6xsin(3x2−2)
This is the derivative of the function f(x) = cos(3x2−2) using the
Chain Rule.
Question 12
Problem: Given the function f(x) = √3x2+ 2x+ 1, use the chain
rule to find the derivative f′(x).
—
Solution:
Step 1: Identify the inner and outer functions. - Let u(x)=3x2+
2x+ 1 be the inner function. - The outer function is g(u) = √u.
Step 2: Differentiate both functions separately. - Differentiate
u(x):
u′(x) = d
dx (3x2+ 2x+ 1) = 6x+ 2
- Differentiate g(u)with respect to u:
g′(u) = d
du (√u) = 1
2√u
Step 3: Apply the chain rule. - According to the chain rule, f′(x) =
g′(u(x)) ·u′(x):
f′(x) = 1
2pu(x)·(6x+ 2)
Step 4: Substitute u(x)back into the derivative. - Replace u(x)
with 3x2+ 2x+ 1:
f′(x) = 1
2√3x2+ 2x+ 1 ·(6x+ 2)
- Simplify the expression:
f′(x) = 6x+ 2
2√3x2+ 2x+ 1
- Further simplification leads to:
f′(x) = 3x+ 1
√3x2+ 2x+ 1
15
Conclusion: The derivative of the function f(x) = √3x2+ 2x+ 1 is:
f′(x) = 3x+ 1
√3x2+ 2x+ 1
Question 12: Differentiation - Chain Rule
Problem: Given the function f(x) = √3x2+ 2x+ 1, use the chain
rule to find the derivative f′(x).
—
Solution:
Step 1: Identify the inner and outer functions. - Let u(x)=3x2+
2x+ 1 be the inner function. - The outer function is g(u) = √u.
Step 2: Differentiate both functions separately. - Differentiate
u(x):
u′(x) = d
dx (3x2+ 2x+ 1) = 6x+ 2
- Differentiate g(u)with respect to u:
g′(u) = d
du (√u) = 1
2√u
Step 3: Apply the chain rule. - According to the chain rule, f′(x) =
g′(u(x)) ·u′(x):
f′(x) = 1
2pu(x)·(6x+ 2)
Step 4: Substitute u(x)back into the derivative. - Replace u(x)
with 3x2+ 2x+ 1:
f′(x) = 1
2√3x2+ 2x+ 1 ·(6x+ 2)
- Simplify the expression:
f′(x) = 6x+ 2
2√3x2+ 2x+ 1
- Further simplification leads to:
f′(x) = 3x+ 1
√3x2+ 2x+ 1
Conclusion: The derivative of the function f(x) = √3x2+ 2x+ 1 is:
f′(x) = 3x+ 1
√3x2+ 2x+ 1
16
Question 13
Problem: Find the derivative of the function f(x) = √3x2+ 2x+ 1.
Solution:
Step 1: Identify the outer and inner functions. - Let g(x) = 3x2+
2x+ 1 (inner function). - Let f(x) = √x(outer function, applied to
g(x)).
Step 2: Differentiate the outer function with respect to the inner
function. - The derivative of f(x) = √xis f′(x) = 1
2√x. - Applying the
chain rule, replace xwith g(x):f′(g(x)) = 1
2√g(x).
Step 3: Differentiate the inner function g(x). - g(x) = 3x2+ 2x+ 1.
- Using the power rule,
g′(x) = d
dx (3x2) + d
dx (2x) + d
dx (1) = 6x+ 2.
Step 4: Apply the chain rule. - Multiply the derivatives from
Steps 2 and 3:
f′(x) = f′(g(x)) ·g′(cardperalf′(x) = 1
2√3x2+ 2x+ 1 ·(6x+ 2).
Step 5: Simplify the expression (if needed). - Combine the deriva-
tives, resulting in
f′(x) = 6x+ 2
2√3x2+ 2x+ 1.
- Simplify further by factoring out 2 from the numerator:
f′(x) = 2(3x+ 1)
2√3x2+ 2x+ 1 =3x+ 1
√3x2+ 2x+ 1.
Conclusion: The derivative of the function f(x) = √3x2+ 2x+ 1
using the chain rule is 3x+1
√3x2+2x+1 . Question 13: Differentiation - Chain
Rule
Problem: Find the derivative of the function f(x) = √3x2+ 2x+ 1.
Solution:
Step 1: Identify the outer and inner functions. - Let g(x) = 3x2+
2x+ 1 (inner function). - Let f(x) = √x(outer function, applied to
g(x)).
Step 2: Differentiate the outer function with respect to the inner
function. - The derivative of f(x) = √xis f′(x) = 1
2√x. - Applying the
chain rule, replace xwith g(x):f′(g(x)) = 1
2√g(x).
Step 3: Differentiate the inner function g(x). - g(x) = 3x2+ 2x+ 1.
- Using the power rule,
g′(x) = d
dx (3x2) + d
dx (2x) + d
dx (1) = 6x+ 2.
17
Step 4: Apply the chain rule. - Multiply the derivatives from
Steps 2 and 3:
f′(x) = f′(g(x)) ·g′(cardperalf′(x) = 1
2√3x2+ 2x+ 1 ·(6x+ 2).
Step 5: Simplify the expression (if needed). - Combine the deriva-
tives, resulting in
f′(x) = 6x+ 2
2√3x2+ 2x+ 1.
- Simplify further by factoring out 2 from the numerator:
f′(x) = 2(3x+ 1)
2√3x2+ 2x+ 1 =3x+ 1
√3x2+ 2x+ 1.
Conclusion: The derivative of the function f(x) = √3x2+ 2x+ 1
using the chain rule is 3x+1
√3x2+2x+1 .
Question 14
—
Consider the function y= cos(3x2−2x+ 1). Use the chain rule to
find the derivative of the function with respect to x.
Solution
Step 1: Identify the Outer and Inner Functions
In the composite function y= cos(3x2−2x+1), identify: - The outer
function, f(u) = cos(u)- The inner function, u= 3x2−2x+ 1
Step 2: Differentiate the Outer Function
Differentiate the outer function f(u)with respect to u:
f′(u) = −sin(u)
Step 3: Differentiate the Inner Function
Differentiate the inner function uwith respect to x:
u′=d
dx (3x2−2x+ 1) = 6x−2
Step 4: Apply the Chain Rule
Apply the chain rule dy
dx =f′(u)·u′:
dy
dx =−sin(3x2−2x+ 1) ·(6x−2)
Step 5: Simplify
Combine and simplify the expression:
dy
dx =−(6x−2) sin(3x2−2x+ 1)
18
Final Answer
The derivative of the function y= cos(3x2−2x+ 1) with respect to
xis: dy
dx =−(6x−2) sin(3x2−2x+ 1)
This concludes the differentiation using the chain rule for this func-
tion. Question 14: Differentiation Using the Chain Rule
—
Consider the function y= cos(3x2−2x+ 1). Use the chain rule to
find the derivative of the function with respect to x.
Solution
Step 1: Identify the Outer and Inner Functions
In the composite function y= cos(3x2−2x+1), identify: - The outer
function, f(u) = cos(u)- The inner function, u= 3x2−2x+ 1
Step 2: Differentiate the Outer Function
Differentiate the outer function f(u)with respect to u:
f′(u) = −sin(u)
Step 3: Differentiate the Inner Function
Differentiate the inner function uwith respect to x:
u′=d
dx (3x2−2x+ 1) = 6x−2
Step 4: Apply the Chain Rule
Apply the chain rule dy
dx =f′(u)·u′:
dy
dx =−sin(3x2−2x+ 1) ·(6x−2)
Step 5: Simplify
Combine and simplify the expression:
dy
dx =−(6x−2) sin(3x2−2x+ 1)
Final Answer
The derivative of the function y= cos(3x2−2x+ 1) with respect to
xis: dy
dx =−(6x−2) sin(3x2−2x+ 1)
This concludes the differentiation using the chain rule for this func-
tion.
19
Question 15
Given the function f(x) = √3x2+ 2x+ 4, use the chain rule to find
f′(x), the derivative of the function with respect to x.
Solution Steps
To find f′(x), the derivative of f(x)with respect to x, follow these
steps:
Step 1: Identify the Outer and Inner Functions. In the given
function f(x) = √3x2+ 2x+ 4:
- The outer function g(u)can be considered as √u. - The inner
function u(x)can be identified as 3x2+ 2x+ 4.
Step 2: Differentiate the Outer Function. The derivative of the
outer function g(u) = √uwith respect to uusing the power rule is:
g′(u) = 1
2u−1/2=1
2√u
Step 3: Differentiate the Inner Function. Differentiate u(x) = 3x2+
2x+ 4 with respect to x:
u′(x) = d
dx (3x2) + d
dx (2x) + d
dx (4)
u′(x) = 6x+ 2 + 0 = 6x+ 2
Step 4: Apply the Chain Rule. The Chain Rule states that if
f(x) = g(u(x)), then:
f′(x) = g′(u(x)) ·u′(x)
Plugging in the derivatives we found:
f′(x) = 1
2√3x2+ 2x+ 4 ·(6x+ 2)
Step 5: Simplify the Expression.
f′(x) = 6x+ 2
2√3x2+ 2x+ 4 =3x+ 1
√3x2+ 2x+ 4
Therefore, the derivative f′(x)of the function f(x) = √3x2+ 2x+ 4
with respect to xis:
f′(x) = 3x+ 1
√3x2+ 2x+ 4
Question 15: Differentiation Using the Chain Rule
Given the function f(x) = √3x2+ 2x+ 4, use the chain rule to find
f′(x), the derivative of the function with respect to x.
Solution Steps
20
To find f′(x), the derivative of f(x)with respect to x, follow these
steps:
Step 1: Identify the Outer and Inner Functions. In the given
function f(x) = √3x2+ 2x+ 4:
- The outer function g(u)can be considered as √u. - The inner
function u(x)can be identified as 3x2+ 2x+ 4.
Step 2: Differentiate the Outer Function. The derivative of the
outer function g(u) = √uwith respect to uusing the power rule is:
g′(u) = 1
2u−1/2=1
2√u
Step 3: Differentiate the Inner Function. Differentiate u(x) = 3x2+
2x+ 4 with respect to x:
u′(x) = d
dx (3x2) + d
dx (2x) + d
dx (4)
u′(x) = 6x+ 2 + 0 = 6x+ 2
Step 4: Apply the Chain Rule. The Chain Rule states that if
f(x) = g(u(x)), then:
f′(x) = g′(u(x)) ·u′(x)
Plugging in the derivatives we found:
f′(x) = 1
2√3x2+ 2x+ 4 ·(6x+ 2)
Step 5: Simplify the Expression.
f′(x) = 6x+ 2
2√3x2+ 2x+ 4 =3x+ 1
√3x2+ 2x+ 4
Therefore, the derivative f′(x)of the function f(x) = √3x2+ 2x+ 4
with respect to xis:
f′(x) = 3x+ 1
√3x2+ 2x+ 4
21
Final Answer:
f′(Quick(x) = 3x+ 1
√3x2+ 2x+ 1
Question 2: Differentiate the function using the Chain Rule
Given the function f(x) = √3x2+ 2x+ 1, find f′(x), the derivative
of the function with respect to x.
—
Solution:
Step 1: Identify the Outer and Inner Functions - The outer func-
tion g(u)is √u- The inner function u(x)is 3x2+ 2x+ 1
Step 2: Differentiate the Outer Function - The derivative of g(u) =
√uwith respect to uis g′(u) = 1
2√u
Step 3: Differentiate the Inner Function - The derivative of u(x) =
3x2+ 2x+ 1 with respect to xis u′(x)=6x+ 2
Step 4: Apply the Chain Rule - The chain rule states that f′(x) =
g′(u)·u′(x). Plug in the derivatives from Step 2 and Step 3:
f′(x) = 1
2√3x2+ 2x+ 1 ·(6x+ 2)
Step 5: Simplify the Expression - Combine and simplify the ex-
pression:
f′(x) = 6x+ 2
2√3x2+ 2x+ 1
- Further simplification gives:
f′(x) = 3x+ 1
√3x2+ 2x+ 1
Final Answer:
f′(Quick(x) = 3x+ 1
√3x2+ 2x+ 1
Question 3
Find the derivative y′of the function y= sin(e2x).
Solution:
Step 1: Recognize the outer function and the inner function. -
Let u=e2x, which is the inner function. - The outer function then
becomes y= sin(u).
Step 2: Differentiate the outer function ywith respect to u. -
Differentiate y= sin(u)with respect to uto get dy
du = cos(u).
Step 3: Differentiate the inner function uwith respect to x. -
Differentiate u=e2xwith respect to xto get du
dx = 2e2x.
3
Step 4: Apply the chain rule. - Multiply the derivatives from Step
2 and Step 3 to get the derivative of ywith respect to x.
y′=dy
dx =dy
du ·du
dx = cos(u)·2e2x
Step 5: Substitute back the expression for u. - Substitute u=e2x
back into the derivative:
y′= 2e2xcos(e2x)
Final Answer: The derivative of y= sin(e2x)is y′= 2e2xcos(e2x).
Question:
Find the derivative y′of the function y= sin(e2x).
Solution:
Step 1: Recognize the outer function and the inner function. -
Let u=e2x, which is the inner function. - The outer function then
becomes y= sin(u).
Step 2: Differentiate the outer function ywith respect to u. -
Differentiate y= sin(u)with respect to uto get dy
du = cos(u).
Step 3: Differentiate the inner function uwith respect to x. -
Differentiate u=e2xwith respect to xto get du
dx = 2e2x.
Step 4: Apply the chain rule. - Multiply the derivatives from Step
2 and Step 3 to get the derivative of ywith respect to x.
y′=dy
dx =dy
du ·du
dx = cos(u)·2e2x
Step 5: Substitute back the expression for u. - Substitute u=e2x
back into the derivative:
y′= 2e2xcos(e2x)
Final Answer: The derivative of y= sin(e2x)is y′= 2e2xcos(e2x).
Question 4
Problem: Differentiate the function f(x) = sin(5x2+ 3x)using the
chain rule.
Solution:
Step 1: Identify the outer and inner functions. - Outer function,
g(u), where u= 5x2+ 3xis sin(u). - Inner function, u(x), is 5x2+ 3x.
Step 2: Differentiate the outer function with respect to the inner
function. - Differentiate g(u) = sin(u)with respect to u:
g′(u) = cos(u)
4
Step 3: Differentiate the inner function with respect to x. - Dif-
ferentiate u(x)=5x2+ 3xwith respect to x:
u′(x) = 10x+ 3
Step 4: Apply the chain rule. - Multiply the derivatives obtained
in steps 2 and 3:
f′(x) = g′(u)·u′(x) = cos(5x2+ 3x)·(10x+ 3)
Step 5: Combine and express the final result. - Thus, the deriva-
tive of f(x)is:
f′(x) = cos(5x2+ 3x)·(10x+ 3)
This completes the differentiation process using the chain rule for
the given function. Question 4: Differentiation - Chain Rule
Problem: Differentiate the function f(x) = sin(5x2+ 3x)using the
chain rule.
Solution:
Step 1: Identify the outer and inner functions. - Outer function,
g(u), where u= 5x2+ 3xis sin(u). - Inner function, u(x), is 5x2+ 3x.
Step 2: Differentiate the outer function with respect to the inner
function. - Differentiate g(u) = sin(u)with respect to u:
g′(u) = cos(u)
Step 3: Differentiate the inner function with respect to x. - Dif-
ferentiate u(x) = 5x2+ 3xwith respect to x:
u′(x) = 10x+ 3
Step 4: Apply the chain rule. - Multiply the derivatives obtained
in steps 2 and 3:
f′(x) = g′(u)·u′(x) = cos(5x2+ 3x)·(10x+ 3)
Step 5: Combine and express the final result. - Thus, the deriva-
tive of f(x)is:
f′(x) = cos(5x2+ 3x)·(10x+ 3)
This completes the differentiation process using the chain rule for
the given function.
Question 5
Step-by-Step Solution:
Step 1: Recognize that the function f(x) = sin2(3x+ 5) can be
written as f(x) = [sin(3x+ 5)]2. Looking at f(x), identify the outer
function and the inner function needed for the chain rule.
5
- Outer function: u2where u= sin(3x+ 5). - Inner function: u=
sin(3x+ 5).
Step 2: Differentiate the outer function with respect to u:
d
du (u2)=2u.
Step 3: Differentiate the inner function with respect to x:
d
dx (sin(3x+ 5)) = cos(3x+ 5) ·d
dx (3x+ 5) = 3 cos(3x+ 5).
Step 4: Apply the chain rule. Multiply the derivative of the outer
function by the derivative of the inner function:
df
dx = 2(sin(3x+ 5)) ·3 cos(3x+ 5) = 6 sin(3x+ 5) cos(3x+ 5).
Step 5: Optionally, use the double angle identity: sin(2θ) = 2 sin(θ) cos(θ),
to simplify the expression wherein θ= 3x+ 5:
6 sin(3x+ 5) cos(3x+ 5) = 3 sin(2(3x+ 5)) = 3 sin(6x+ 10).
Final Answer:
df
dx = 3 sin(6x+ 10).
Question 5: Differentiate f(x) = sin2(3x+ 5).
Step-by-Step Solution:
Step 1: Recognize that the function f(x) = sin2(3x+ 5) can be
written as f(x) = [sin(3x+ 5)]2. Looking at f(x), identify the outer
function and the inner function needed for the chain rule.
- Outer function: u2where u= sin(3x+ 5). - Inner function: u=
sin(3x+ 5).
Step 2: Differentiate the outer function with respect to u:
d
du (u2)=2u.
Step 3: Differentiate the inner function with respect to x:
d
dx (sin(3x+ 5)) = cos(3x+ 5) ·d
dx (3x+ 5) = 3 cos(3x+ 5).
Step 4: Apply the chain rule. Multiply the derivative of the outer
function by the derivative of the inner function:
df
dx = 2(sin(3x+ 5)) ·3 cos(3x+ 5) = 6 sin(3x+ 5) cos(3x+ 5).
6
Step 5: Optionally, use the double angle identity: sin(2θ) = 2 sin(θ) cos(θ),
to simplify the expression wherein θ= 3x+ 5:
6 sin(3x+ 5) cos(3x+ 5) = 3 sin(2(3x+ 5)) = 3 sin(6x+ 10).
Final Answer:
df
dx = 3 sin(6x+ 10).
Question 6
Problem Statement: Find the derivative of the function f(x) =
sin(3x2+ 2x)using the chain rule.
—
Solution:
Step 1: Identify the outer and inner functions. - The outer func-
tion is g(u) = sin(u). - The inner function is h(x) = 3x2+ 2x.
Step 2: Differentiate the outer function with respect to the inner
function. - The derivative of the outer function g(u)with respect to
uis g′(u) = cos(u).
Step 3: Differentiate the inner function. - The derivative of h(x) =
3x2+ 2xwith respect to xis h′(x) = 6x+ 2 (using basic differentiation
rules).
Step 4: Apply the chain rule. - The chain rule states that (g◦
h)′(x) = g′(h(x)) ×h′(x). - Plugging in the derivatives you found, you
get:
f′(x) = cos(3x2+ 2x)·(6x+ 2)
Step 5: Simplify the expression. - The simplified form of the
derivative is:
f′(x) = (6x+ 2) cos(3x2+ 2x)
Final Answer: The derivative of the function f(x) = sin(3x2+ 2x)
is f′(x) = (6x+ 2) cos(3x2+ 2x). Question 6: Differentiation Using the
Chain Rule
Problem Statement: Find the derivative of the function f(x) =
sin(3x2+ 2x)using the chain rule.
—
Solution:
Step 1: Identify the outer and inner functions. - The outer func-
tion is g(u) = sin(u). - The inner function is h(x)=3x2+ 2x.
Step 2: Differentiate the outer function with respect to the inner
function. - The derivative of the outer function g(u)with respect to
uis g′(u) = cos(u).
Step 3: Differentiate the inner function. - The derivative of h(x) =
3x2+ 2xwith respect to xis h′(x) = 6x+ 2 (using basic differentiation
rules).
7
Step 4: Apply the chain rule. - The chain rule states that (g◦
h)′(x) = g′(h(x)) ×h′(x). - Plugging in the derivatives you found, you
get:
f′(x) = cos(3x2+ 2x)·(6x+ 2)
Step 5: Simplify the expression. - The simplified form of the
derivative is:
f′(x) = (6x+ 2) cos(3x2+ 2x)
Final Answer: The derivative of the function f(x) = sin(3x2+ 2x)is
f′(x) = (6x+ 2) cos(3x2+ 2x).
Question 7
Given the function f(x) = sin(5x2+ 3x−2), use the chain rule to
find the derivative f′(x).
Solution:
Step 1: Identify the Outer and Inner Functions - Outer function:
g(u) = sin(u), where uis the function inside the parentheses. - Inner
function: h(x) = 5x2+ 3x−2, which is the expression inside the sine
function.
Step 2: Differentiate the Outer Function - Differentiate g(u)with
respect to uto get g′(u) = cos(u).
Step 3: Differentiate the Inner Function - Differentiate h(x) =
5x2+ 3x−2with respect to x.
h′(x) = (5 ·2x)+3−0 = 10x+ 3
Step 4: Apply the Chain Rule - The chain rule states that f′(x) =
g′(h(x))·h′(x). - Substitute g′(h(x)) = cos(5x2+3x−2) and h′(x) = 10x+3.
Step 5: Write the Final Result - Combine the derivatives:
f′(x) = cos(5x2+ 3x−2) ·(10x+ 3)
This is the derivative of f(x)using the chain rule.
Additional Instruction Question:
Verify f′(x)by evaluating f′(2) using your result.
Solution:
Step 1: Substitute x= 2 into f′(x)- Replace xwith 2 in the deriva-
tive:
f′(2) = cos(5(2)2+ 3(2) −2) ·(10(2) + 3)
Step 2: Simplify the Calculation - Calculate inside the cosine and
then the multiplication:
f′(2) = cos(5 ·4+6−2) ·(20 + 3) = cos(20 + 4) ·23 = cos(24) ·23
8
Step 3: Numeric Evaluation (if a calculator is allowed) - Evaluate
cos(24) using a calculator and multiply by 23 to get the numerical
value.
f′(2) cos(24◦)·23
(Here, you might need a calculator to find the precise value, or you
can keep it in this symbolic form if the problem does not require a
numerical answer.)
These steps show how to differentiate a function using the chain
rule and to verify your result through a specific substitution. Ques-
tion 7: Differentiation Using the Chain Rule
Given the function f(x) = sin(5x2+ 3x−2), use the chain rule to
find the derivative f′(x).
Solution:
Step 1: Identify the Outer and Inner Functions - Outer function:
g(u) = sin(u), where uis the function inside the parentheses. - Inner
function: h(x) = 5x2+ 3x−2, which is the expression inside the sine
function.
Step 2: Differentiate the Outer Function - Differentiate g(u)with
respect to uto get g′(u) = cos(u).
Step 3: Differentiate the Inner Function - Differentiate h(x) =
5x2+ 3x−2with respect to x.
h′(x) = (5 ·2x)+3−0 = 10x+ 3
Step 4: Apply the Chain Rule - The chain rule states that f′(x) =
g′(h(x))·h′(x). - Substitute g′(h(x)) = cos(5x2+3x−2) and h′(x) = 10x+3.
Step 5: Write the Final Result - Combine the derivatives:
f′(x) = cos(5x2+ 3x−2) ·(10x+ 3)
This is the derivative of f(x)using the chain rule.
Additional Instruction Question:
Verify f′(x)by evaluating f′(2) using your result.
Solution:
Step 1: Substitute x= 2 into f′(x)- Replace xwith 2 in the deriva-
tive:
f′(2) = cos(5(2)2+ 3(2) −2) ·(10(2) + 3)
Step 2: Simplify the Calculation - Calculate inside the cosine and
then the multiplication:
f′(2) = cos(5 ·4+6−2) ·(20 + 3) = cos(20 + 4) ·23 = cos(24) ·23
Step 3: Numeric Evaluation (if a calculator is allowed) - Evaluate
cos(24) using a calculator and multiply by 23 to get the numerical
value.
f′(2) cos(24◦)·23
9
(Here, you might need a calculator to find the precise value, or you
can keep it in this symbolic form if the problem does not require a
numerical answer.)
These steps show how to differentiate a function using the chain
rule and to verify your result through a specific substitution.
Question 8
Step-by-Step Solution:
Step 1: Identify the outer and inner functions. - The outer func-
tion is g(u) = eu. - The inner function is u(x)=3x2+ 2x.
Step 2: Differentiate the outer function with respect to the inner
function. - The derivative of g(u) = euwith respect to uis g′(u) = eu.
Step 3: Differentiate the inner function with respect to x. - The
derivative of u(x)=3x2+ 2xwith respect to xis u′(x)=6x+ 2.
Step 4: Apply the chain rule. - The chain rule states that f′(x) =
g′(u(x)) ·u′(x).
Step 5: Substitute back the derivatives from steps 2 and 3. -
f′(x) = eu(x)·(6x+ 2).
Step 6: Substitute u(x)=3x2+ 2xback in. - f′(x) = e3x2+2x·(6x+ 2).
Answer: The derivative of the function f(x) = e3x2+2xis f′(x) =
(6x+ 2)e3x2+2x. Question: Determine the derivative of the function
f(x) = e3x2+2xusing the chain rule.
Step-by-Step Solution:
Step 1: Identify the outer and inner functions. - The outer func-
tion is g(u) = eu. - The inner function is u(x)=3x2+ 2x.
Step 2: Differentiate the outer function with respect to the inner
function. - The derivative of g(u) = euwith respect to uis g′(u) = eu.
Step 3: Differentiate the inner function with respect to x. - The
derivative of u(x)=3x2+ 2xwith respect to xis u′(x)=6x+ 2.
Step 4: Apply the chain rule. - The chain rule states that f′(x) =
g′(u(x)) ·u′(x).
Step 5: Substitute back the derivatives from steps 2 and 3. -
f′(x) = eu(x)·(6x+ 2).
Step 6: Substitute u(x)=3x2+ 2xback in. - f′(x) = e3x2+2x·(6x+ 2).
Answer: The derivative of the function f(x) = e3x2+2xis f′(x) =
(6x+ 2)e3x2+2x.
Question 9
Given the function f(x) = √5x2+ 4x−1, find the derivative f′(x).
Solution:
To find the derivative of the function f(x) = √5x2+ 4x−1, we will
use the chain rule. The chain rule states that the derivative of a
10
composite function is the derivative of the outer function evaluated at
the inner function, multiplied by the derivative of the inner function.
Step 1: Identify the inner and outer functions. - Inner function
u(x)=5x2+ 4x−1- Outer function g(u) = √u
Step 2: Differentiate the outer function with respect to the inner
function (u).
g(u) = √u
g′(u) = 1
2√u
Step 3: Differentiate the inner function with respect to x.
u(x) = 5x2+ 4x−1
u′(x) = 10x+ 4
Step 4: Apply the chain rule.
f′(x) = g′(u)·u′(x)
Substitute for g′(u)and u′(x):
f′(x) = 1
2√5x2+ 4x−1·(10x+ 4)
Step 5: Simplify the expression.
f′(x) = 10x+ 4
2√5x2+ 4x−1
Final answer:
f′(x) = 5x+ 2
√5x2+ 4x−1
This is the derivative of the function f(x) = √5x2+ 4x−1using the
chain rule. Question 9 - Differentiation Using the Chain Rule
Given the function f(x) = √5x2+ 4x−1, find the derivative f′(x).
Solution:
To find the derivative of the function f(x) = √5x2+ 4x−1, we will
use the chain rule. The chain rule states that the derivative of a
composite function is the derivative of the outer function evaluated at
the inner function, multiplied by the derivative of the inner function.
Step 1: Identify the inner and outer functions. - Inner function
u(x) = 5x2+ 4x−1- Outer function g(u) = √u
Step 2: Differentiate the outer function with respect to the inner
function (u).
g(u) = √u
g′(u) = 1
2√u
11
Step 3: Differentiate the inner function with respect to x.
u(x) = 5x2+ 4x−1
u′(x) = 10x+ 4
Step 4: Apply the chain rule.
f′(x) = g′(u)·u′(x)
Substitute for g′(u)and u′(x):
f′(x) = 1
2√5x2+ 4x−1·(10x+ 4)
Step 5: Simplify the expression.
f′(x) = 10x+ 4
2√5x2+ 4x−1
Final answer:
f′(x) = 5x+ 2
√5x2+ 4x−1
This is the derivative of the function f(x) = √5x2+ 4x−1using the
chain rule.
Question 10
Find the derivative of the function y=√3x4−5x2+ 2 using the
chain rule.
Solution
Step 1: Identify the outer and inner functions. - The outer func-
tion f(u)is √u. - The inner function g(x)is 3x4−5x2+ 2.
Step 2: Differentiate the outer function with respect to the inner
function. The derivative of f(u) = √uis f′(u) = 1
2√u.
Step 3: Differentiate the inner function with respect to x. The
derivative of g(x)=3x4−5x2+ 2 is:
g′(x) = 12x3−10x
Step 4: Apply the chain rule. The chain rule states that the
derivative of f(g(x)) is f′(g(x)) ·g′(x). Thus, applying the chain rule
gives:
y′=1
2√3x4−5x2+ 2 ·(12x3−10x)
Step 5: Simplify (if possible). To simplify, we can factor out the
common terms in the derivative of the inner function:
y′=1
2√3x4−5x2+ 2 ·2x(6x2−5)
12
y′=x(6x2−5)
√3x4−5x2+ 2
This is the derivative of the function using the chain rule. Question
10: Differentiation - Chain Rule
Find the derivative of the function y=√3x4−5x2+ 2 using the
chain rule.
Solution
Step 1: Identify the outer and inner functions. - The outer func-
tion f(u)is √u. - The inner function g(x)is 3x4−5x2+ 2.
Step 2: Differentiate the outer function with respect to the inner
function. The derivative of f(u) = √uis f′(u) = 1
2√u.
Step 3: Differentiate the inner function with respect to x. The
derivative of g(x)=3x4−5x2+ 2 is:
g′(x) = 12x3−10x
Step 4: Apply the chain rule. The chain rule states that the
derivative of f(g(x)) is f′(g(x)) ·g′(x). Thus, applying the chain rule
gives:
y′=1
2√3x4−5x2+ 2 ·(12x3−10x)
Step 5: Simplify (if possible). To simplify, we can factor out the
common terms in the derivative of the inner function:
y′=1
2√3x4−5x2+ 2 ·2x(6x2−5)
y′=x(6x2−5)
√3x4−5x2+ 2
This is the derivative of the function using the chain rule.
Question 11
Question: Find the derivative of the function f(x) = cos(3x2−2)
using the Chain Rule.
Solution:
Step 1: Identify the outer and inner functions.
The given function is f(x) = cos(3x2−2). Here, the outer function
is cos(u)where u= 3x2−2is the inner function.
Step 2: Differentiate the outer function with respect to the inner
function.
Let u= 3x2−2. The derivative of cos(u)with respect to uis −sin(u).
Step 3: Differentiate the inner function with respect to x.
The derivative of u= 3x2−2with respect to xis du
dx = 6x.
Step 4: Apply the Chain Rule.
13
The Chain Rule states that if a function yis composed of two func-
tions uand g(x)such that y=f(u)and u=g(x), then the derivative
of ywith respect to xis:
dy
dx =df
du ·du
dx
Plugging in the derivatives:
df
dx =−sin(u)·6x
Step 5: Substitute back the original function of u.
Remember, u= 3x2−2. So, substituting back:
df
dx =−sin(3x2−2) ·6x
Final Answer:
df
dx =−6xsin(3x2−2)
This is the derivative of the function f(x) = cos(3x2−2) using the
Chain Rule. Question 11: Differentiation - Chain Rule at Liberty
University
Question: Find the derivative of the function f(x) = cos(3x2−2)
using the Chain Rule.
Solution:
Step 1: Identify the outer and inner functions.
The given function is f(x) = cos(3x2−2). Here, the outer function
is cos(u)where u= 3x2−2is the inner function.
Step 2: Differentiate the outer function with respect to the inner
function.
Let u= 3x2−2. The derivative of cos(u)with respect to uis −sin(u).
Step 3: Differentiate the inner function with respect to x.
The derivative of u= 3x2−2with respect to xis du
dx = 6x.
Step 4: Apply the Chain Rule.
The Chain Rule states that if a function yis composed of two func-
tions uand g(x)such that y=f(u)and u=g(x), then the derivative
of ywith respect to xis:
dy
dx =df
du ·du
dx
Plugging in the derivatives:
df
dx =−sin(u)·6x
Step 5: Substitute back the original function of u.
Remember, u= 3x2−2. So, substituting back:
14
df
dx =−sin(3x2−2) ·6x
Final Answer:
df
dx =−6xsin(3x2−2)
This is the derivative of the function f(x) = cos(3x2−2) using the
Chain Rule.
Question 12
Problem: Given the function f(x) = √3x2+ 2x+ 1, use the chain
rule to find the derivative f′(x).
—
Solution:
Step 1: Identify the inner and outer functions. - Let u(x)=3x2+
2x+ 1 be the inner function. - The outer function is g(u) = √u.
Step 2: Differentiate both functions separately. - Differentiate
u(x):
u′(x) = d
dx (3x2+ 2x+ 1) = 6x+ 2
- Differentiate g(u)with respect to u:
g′(u) = d
du (√u) = 1
2√u
Step 3: Apply the chain rule. - According to the chain rule, f′(x) =
g′(u(x)) ·u′(x):
f′(x) = 1
2pu(x)·(6x+ 2)
Step 4: Substitute u(x)back into the derivative. - Replace u(x)
with 3x2+ 2x+ 1:
f′(x) = 1
2√3x2+ 2x+ 1 ·(6x+ 2)
- Simplify the expression:
f′(x) = 6x+ 2
2√3x2+ 2x+ 1
- Further simplification leads to:
f′(x) = 3x+ 1
√3x2+ 2x+ 1
15
Conclusion: The derivative of the function f(x) = √3x2+ 2x+ 1 is:
f′(x) = 3x+ 1
√3x2+ 2x+ 1
Question 12: Differentiation - Chain Rule
Problem: Given the function f(x) = √3x2+ 2x+ 1, use the chain
rule to find the derivative f′(x).
—
Solution:
Step 1: Identify the inner and outer functions. - Let u(x)=3x2+
2x+ 1 be the inner function. - The outer function is g(u) = √u.
Step 2: Differentiate both functions separately. - Differentiate
u(x):
u′(x) = d
dx (3x2+ 2x+ 1) = 6x+ 2
- Differentiate g(u)with respect to u:
g′(u) = d
du (√u) = 1
2√u
Step 3: Apply the chain rule. - According to the chain rule, f′(x) =
g′(u(x)) ·u′(x):
f′(x) = 1
2pu(x)·(6x+ 2)
Step 4: Substitute u(x)back into the derivative. - Replace u(x)
with 3x2+ 2x+ 1:
f′(x) = 1
2√3x2+ 2x+ 1 ·(6x+ 2)
- Simplify the expression:
f′(x) = 6x+ 2
2√3x2+ 2x+ 1
- Further simplification leads to:
f′(x) = 3x+ 1
√3x2+ 2x+ 1
Conclusion: The derivative of the function f(x) = √3x2+ 2x+ 1 is:
f′(x) = 3x+ 1
√3x2+ 2x+ 1
16
Question 13
Problem: Find the derivative of the function f(x) = √3x2+ 2x+ 1.
Solution:
Step 1: Identify the outer and inner functions. - Let g(x) = 3x2+
2x+ 1 (inner function). - Let f(x) = √x(outer function, applied to
g(x)).
Step 2: Differentiate the outer function with respect to the inner
function. - The derivative of f(x) = √xis f′(x) = 1
2√x. - Applying the
chain rule, replace xwith g(x):f′(g(x)) = 1
2√g(x).
Step 3: Differentiate the inner function g(x). - g(x) = 3x2+ 2x+ 1.
- Using the power rule,
g′(x) = d
dx (3x2) + d
dx (2x) + d
dx (1) = 6x+ 2.
Step 4: Apply the chain rule. - Multiply the derivatives from
Steps 2 and 3:
f′(x) = f′(g(x)) ·g′(cardperalf′(x) = 1
2√3x2+ 2x+ 1 ·(6x+ 2).
Step 5: Simplify the expression (if needed). - Combine the deriva-
tives, resulting in
f′(x) = 6x+ 2
2√3x2+ 2x+ 1.
- Simplify further by factoring out 2 from the numerator:
f′(x) = 2(3x+ 1)
2√3x2+ 2x+ 1 =3x+ 1
√3x2+ 2x+ 1.
Conclusion: The derivative of the function f(x) = √3x2+ 2x+ 1
using the chain rule is 3x+1
√3x2+2x+1 . Question 13: Differentiation - Chain
Rule
Problem: Find the derivative of the function f(x) = √3x2+ 2x+ 1.
Solution:
Step 1: Identify the outer and inner functions. - Let g(x) = 3x2+
2x+ 1 (inner function). - Let f(x) = √x(outer function, applied to
g(x)).
Step 2: Differentiate the outer function with respect to the inner
function. - The derivative of f(x) = √xis f′(x) = 1
2√x. - Applying the
chain rule, replace xwith g(x):f′(g(x)) = 1
2√g(x).
Step 3: Differentiate the inner function g(x). - g(x) = 3x2+ 2x+ 1.
- Using the power rule,
g′(x) = d
dx (3x2) + d
dx (2x) + d
dx (1) = 6x+ 2.
17
Step 4: Apply the chain rule. - Multiply the derivatives from
Steps 2 and 3:
f′(x) = f′(g(x)) ·g′(cardperalf′(x) = 1
2√3x2+ 2x+ 1 ·(6x+ 2).
Step 5: Simplify the expression (if needed). - Combine the deriva-
tives, resulting in
f′(x) = 6x+ 2
2√3x2+ 2x+ 1.
- Simplify further by factoring out 2 from the numerator:
f′(x) = 2(3x+ 1)
2√3x2+ 2x+ 1 =3x+ 1
√3x2+ 2x+ 1.
Conclusion: The derivative of the function f(x) = √3x2+ 2x+ 1
using the chain rule is 3x+1
√3x2+2x+1 .
Question 14
—
Consider the function y= cos(3x2−2x+ 1). Use the chain rule to
find the derivative of the function with respect to x.
Solution
Step 1: Identify the Outer and Inner Functions
In the composite function y= cos(3x2−2x+1), identify: - The outer
function, f(u) = cos(u)- The inner function, u= 3x2−2x+ 1
Step 2: Differentiate the Outer Function
Differentiate the outer function f(u)with respect to u:
f′(u) = −sin(u)
Step 3: Differentiate the Inner Function
Differentiate the inner function uwith respect to x:
u′=d
dx (3x2−2x+ 1) = 6x−2
Step 4: Apply the Chain Rule
Apply the chain rule dy
dx =f′(u)·u′:
dy
dx =−sin(3x2−2x+ 1) ·(6x−2)
Step 5: Simplify
Combine and simplify the expression:
dy
dx =−(6x−2) sin(3x2−2x+ 1)
18
Final Answer
The derivative of the function y= cos(3x2−2x+ 1) with respect to
xis: dy
dx =−(6x−2) sin(3x2−2x+ 1)
This concludes the differentiation using the chain rule for this func-
tion. Question 14: Differentiation Using the Chain Rule
—
Consider the function y= cos(3x2−2x+ 1). Use the chain rule to
find the derivative of the function with respect to x.
Solution
Step 1: Identify the Outer and Inner Functions
In the composite function y= cos(3x2−2x+1), identify: - The outer
function, f(u) = cos(u)- The inner function, u= 3x2−2x+ 1
Step 2: Differentiate the Outer Function
Differentiate the outer function f(u)with respect to u:
f′(u) = −sin(u)
Step 3: Differentiate the Inner Function
Differentiate the inner function uwith respect to x:
u′=d
dx (3x2−2x+ 1) = 6x−2
Step 4: Apply the Chain Rule
Apply the chain rule dy
dx =f′(u)·u′:
dy
dx =−sin(3x2−2x+ 1) ·(6x−2)
Step 5: Simplify
Combine and simplify the expression:
dy
dx =−(6x−2) sin(3x2−2x+ 1)
Final Answer
The derivative of the function y= cos(3x2−2x+ 1) with respect to
xis: dy
dx =−(6x−2) sin(3x2−2x+ 1)
This concludes the differentiation using the chain rule for this func-
tion.
19
Question 15
Given the function f(x) = √3x2+ 2x+ 4, use the chain rule to find
f′(x), the derivative of the function with respect to x.
Solution Steps
To find f′(x), the derivative of f(x)with respect to x, follow these
steps:
Step 1: Identify the Outer and Inner Functions. In the given
function f(x) = √3x2+ 2x+ 4:
- The outer function g(u)can be considered as √u. - The inner
function u(x)can be identified as 3x2+ 2x+ 4.
Step 2: Differentiate the Outer Function. The derivative of the
outer function g(u) = √uwith respect to uusing the power rule is:
g′(u) = 1
2u−1/2=1
2√u
Step 3: Differentiate the Inner Function. Differentiate u(x) = 3x2+
2x+ 4 with respect to x:
u′(x) = d
dx (3x2) + d
dx (2x) + d
dx (4)
u′(x) = 6x+ 2 + 0 = 6x+ 2
Step 4: Apply the Chain Rule. The Chain Rule states that if
f(x) = g(u(x)), then:
f′(x) = g′(u(x)) ·u′(x)
Plugging in the derivatives we found:
f′(x) = 1
2√3x2+ 2x+ 4 ·(6x+ 2)
Step 5: Simplify the Expression.
f′(x) = 6x+ 2
2√3x2+ 2x+ 4 =3x+ 1
√3x2+ 2x+ 4
Therefore, the derivative f′(x)of the function f(x) = √3x2+ 2x+ 4
with respect to xis:
f′(x) = 3x+ 1
√3x2+ 2x+ 4
Question 15: Differentiation Using the Chain Rule
Given the function f(x) = √3x2+ 2x+ 4, use the chain rule to find
f′(x), the derivative of the function with respect to x.
Solution Steps
20
To find f′(x), the derivative of f(x)with respect to x, follow these
steps:
Step 1: Identify the Outer and Inner Functions. In the given
function f(x) = √3x2+ 2x+ 4:
- The outer function g(u)can be considered as √u. - The inner
function u(x)can be identified as 3x2+ 2x+ 4.
Step 2: Differentiate the Outer Function. The derivative of the
outer function g(u) = √uwith respect to uusing the power rule is:
g′(u) = 1
2u−1/2=1
2√u
Step 3: Differentiate the Inner Function. Differentiate u(x) = 3x2+
2x+ 4 with respect to x:
u′(x) = d
dx (3x2) + d
dx (2x) + d
dx (4)
u′(x) = 6x+ 2 + 0 = 6x+ 2
Step 4: Apply the Chain Rule. The Chain Rule states that if
f(x) = g(u(x)), then:
f′(x) = g′(u(x)) ·u′(x)
Plugging in the derivatives we found:
f′(x) = 1
2√3x2+ 2x+ 4 ·(6x+ 2)
Step 5: Simplify the Expression.
f′(x) = 6x+ 2
2√3x2+ 2x+ 4 =3x+ 1
√3x2+ 2x+ 4
Therefore, the derivative f′(x)of the function f(x) = √3x2+ 2x+ 4
with respect to xis:
f′(x) = 3x+ 1
√3x2+ 2x+ 4
21
