MATH 332 - ADVANCED CALCULUS -
Implicit Differentiation Question Bank
Question 1
Solution: Given equation is x2+y2= 5.
1. Take the derivative of both sides with respect to x:
d
dx (x2) + d
dx (y2) = d
dx (5)
2. Use chain rule for d
dx (y2):
2x+ 2ydy
dx = 0
3. Solve for dy
dx :
2ydy
dx =−2x
dy
dx =
−2x
2y=
−x
y
Therefore, dy
dx =−x
y.Question 1: Find dy
dx for the equation x2+y2= 5
using implicit differentiation.
Solution: Given equation is x2+y2= 5.
1. Take the derivative of both sides with respect to x:
d
dx (x2) + d
dx (y2) = d
dx (5)
2. Use chain rule for d
dx (y2):
2x+ 2ydy
dx = 0
3. Solve for dy
dx :
2ydy
dx =−2x
dy
dx =
−2x
2y=
−x
y
Therefore, dy
dx =−x
y.
1
Question 2
Find dy
dx by implicit differentiation: x3+y3= 9xy.
Solution:
Given equation: x3+y3= 9xy
Taking derivative of both sides with respect to x:
d
dx (x3) + d
dx (y3) = d
dx (9xy)
3x2+ 3y2dy
dx = 9y+ 9xdy
dx
Moving terms involving dy
dx to one side:
3y2dy
dx
−9xdy
dx = 9y−3x2
Factor out dy
dx :
dy
dx (3y2
−9x)=9y−3x2
Now, solve for dy
dx :
dy
dx =9y−3x2
3y2−9x
Therefore, dy
dx =9y−3x2
3y2−9xQuestion 2:
Find dy
dx by implicit differentiation: x3+y3= 9xy.
Solution:
Given equation: x3+y3= 9xy
Taking derivative of both sides with respect to x:
d
dx (x3) + d
dx (y3) = d
dx (9xy)
3x2+ 3y2dy
dx = 9y+ 9xdy
dx
Moving terms involving dy
dx to one side:
3y2dy
dx
−9xdy
dx = 9y−3x2
Factor out dy
dx :
dy
dx (3y2
−9x)=9y−3x2
Now, solve for dy
dx :
2
dy
dx =9y−3x2
3y2−9x
Therefore, dy
dx =9y−3x2
3y2−9x
Question 3
Step-by-step Solution: Given equation: 2x2+ 3y2= 5xy
1. Differentiate both sides of the equation with respect to xusing
the rules of implicit differentiation.
d
dx (2x2) + d
dx (3y2) = d
dx (5xy)
2·2x+ 3 ·2ydy
dx = 5xdy
dx + 5y
2. Simplify the equation by isolating dy
dx on one side.
4x+ 6ydy
dx = 5xdy
dx + 5y
6ydy
dx
−5xdy
dx = 5y−4x
dy
dx (6y−5x)=5y−4x
dy
dx =5y−4x
6y−5x
Therefore, the derivative of ywith respect to xis 5y−4x
6y−5x.Question
3: Find dy
dx by implicit differentiation: 2x2+ 3y2= 5xy
Step-by-step Solution: Given equation: 2x2+ 3y2= 5xy
1. Differentiate both sides of the equation with respect to xusing
the rules of implicit differentiation.
d
dx (2x2) + d
dx (3y2) = d
dx (5xy)
2·2x+ 3 ·2ydy
dx = 5xdy
dx + 5y
2. Simplify the equation by isolating dy
dx on one side.
4x+ 6ydy
dx = 5xdy
dx + 5y
6ydy
dx
−5xdy
dx = 5y−4x
dy
dx (6y−5x)=5y−4x
dy
dx =5y−4x
6y−5x
3
Therefore, the derivative of ywith respect to xis 5y−4x
6y−5x.
Question 4
Step-by-step solution: Step 1: Differentiate both sides with re-
spect to x:
d
dx (x2) + d
dx (y2) = d
dx (4)
Step 2: Simplify using the chain rule for d
dx (y2)and the constant
rule for d
dx (4):
2x+ 2ydy
dx = 0
Step 3: Solve for dy
dx :
2ydy
dx =−2x
dy
dx =
−2x
2y=
−x
y
Therefore, the derivative of ywith respect to xis −x
y.Question 4:
Find dy
dx by implicit differentiation: x2+y2= 4.
Step-by-step solution: Step 1: Differentiate both sides with re-
spect to x:
d
dx (x2) + d
dx (y2) = d
dx (4)
Step 2: Simplify using the chain rule for d
dx (y2)and the constant
rule for d
dx (4):
2x+ 2ydy
dx = 0
Step 3: Solve for dy
dx :
2ydy
dx =−2x
dy
dx =
−2x
2y=
−x
y
Therefore, the derivative of ywith respect to xis −x
y.
Question 5
Find dy
dx by implicit differentiation:
3x2+ 4xy + 2y2= 12
Solution:
4
To find dy
dx by implicit differentiation, follow these steps:
Step 1: Differentiate both sides of the equation with respect to x:
d
dx (3x2) + d
dx (4xy) + d
dx (2y2) = d
dx (12)
Step 2: Simplify each term using the chain rule and product rule
as needed:
6x+ 4xdy
dx + 4y+ 4ydy
dx = 0
Step 3: Rearrange the terms to solve for dy
dx :
4xdy
dx + 4ydy
dx =−6x−4y
Step 4: Factor out dy
dx :
dy
dx (4x+ 4y) = −6x−4y
Step 5: Solve for dy
dx :
dy
dx =
−6x−4y
4x+ 4y
Question 5:
Find dy
dx by implicit differentiation:
3x2+ 4xy + 2y2= 12
Solution:
To find dy
dx by implicit differentiation, follow these steps:
Step 1: Differentiate both sides of the equation with respect to x:
d
dx (3x2) + d
dx (4xy) + d
dx (2y2) = d
dx (12)
Step 2: Simplify each term using the chain rule and product rule
as needed:
6x+ 4xdy
dx + 4y+ 4ydy
dx = 0
Step 3: Rearrange the terms to solve for dy
dx :
4xdy
dx + 4ydy
dx =−6x−4y
Step 4: Factor out dy
dx :
dy
dx (4x+ 4y) = −6x−4y
5
Step 5: Solve for dy
dx :
dy
dx =
−6x−4y
4x+ 4y
Question 6
Solution: Given equation: x2+y2= 1
Step 1: Differentiate both sides of the equation with respect to x
using implicit differentiation.
d
dx (x2) + d
dx (y2) = d
dx (1)
2x+ 2ydy
dx = 0
Step 2: Solve for dy
dx .
2x+ 2ydy
dx = 0
2ydy
dx =−2x
dy
dx =
−2x
2y
dy
dx =
−x
y
Therefore, dy
dx =−x
y.Question 6: Find dy
dx for x2+y2= 1 using implicit
differentiation.
Solution: Given equation: x2+y2= 1
Step 1: Differentiate both sides of the equation with respect to x
using implicit differentiation.
d
dx (x2) + d
dx (y2) = d
dx (1)
2x+ 2ydy
dx = 0
Step 2: Solve for dy
dx .
2x+ 2ydy
dx = 0
2ydy
dx =−2x
dy
dx =
−2x
2y
dy
dx =
−x
y
Therefore, dy
dx =−x
y.
6
Question 7
Step-by-step solution: To find dy
dx using implicit differentiation, we
follow these steps:
1. Differentiate both sides of the equation with respect to x. 2.
Apply the chain rule when differentiating terms involving y. 3. Solve
for dy
dx .
Given equation: x2+y2= 5
Step 1: Differentiate both sides with respect to x:
d
dx (x2) + d
dx (y2) = d
dx (5)
2x+ 2ydy
dx = 0
Step 2: Apply the chain rule to the term 2ydy
dx :
2x+ 2yd
dx (dy
dx )=0
2x+ 2yy′= 0
Step 3: Solve for dy
dx :
2yy′=−2x
y′=
−2x
2y
dy
dx =
−x
y
Therefore, the derivative of ywith respect to xis −x
y.Question 7:
Find dy
dx for the equation x2+y2= 5 using implicit differentiation.
Step-by-step solution: To find dy
dx using implicit differentiation, we
follow these steps:
1. Differentiate both sides of the equation with respect to x. 2.
Apply the chain rule when differentiating terms involving y. 3. Solve
for dy
dx .
Given equation: x2+y2= 5
Step 1: Differentiate both sides with respect to x:
d
dx (x2) + d
dx (y2) = d
dx (5)
2x+ 2ydy
dx = 0
Step 2: Apply the chain rule to the term 2ydy
dx :
2x+ 2yd
dx (dy
dx )=0
7
2x+ 2yy′= 0
Step 3: Solve for dy
dx :
2yy′=−2x
y′=
−2x
2y
dy
dx =
−x
y
Therefore, the derivative of ywith respect to xis −x
y.
Question 8
Step-by-step solution: 1. Differentiate both sides of the equation
with respect to x. 2. For x2, apply power rule to get 2x. 3. For
y2, differentiate implicitly using the chain rule to get 2ydy
dx . 4. The
derivative of a constant (9) with respect to xis 0. 5. Combine the
results to get 2x+ 2ydy
dx = 0. 6. Solve for dy
dx by isolating it: dy
dx =−x
y.
Therefore, dy
dx =−x
y.Question 8: Find dy
dx by implicit differentiation:
x2+y2= 9
Step-by-step solution: 1. Differentiate both sides of the equation
with respect to x. 2. For x2, apply power rule to get 2x. 3. For
y2, differentiate implicitly using the chain rule to get 2ydy
dx . 4. The
derivative of a constant (9) with respect to xis 0. 5. Combine the
results to get 2x+ 2ydy
dx = 0. 6. Solve for dy
dx by isolating it: dy
dx =−x
y.
Therefore, dy
dx =−x
y.
Question 9
Find dy
dx by implicit differentiation.
x2y+y2= 6x
Solution:
1. Differentiate both sides of the equation with respect to xusing
implicit differentiation.
d
dx (x2y+y2) = d
dx (6x)
d
dx (x2y) + d
dx (y2)=6
8
2. Apply the product rule for differentiation of x2yand the chain
rule for differentiation of y2.
2xy +x2dy
dx + 2ydy
dx = 6
x2dy
dx + 2ydy
dx = 6 −2xy
dy
dx (x2+ 2y)=6−2xy
dy
dx =6−2xy
x2+ 2y
Therefore, dy
dx =6−2xy
x2+2y.Question 9:
Find dy
dx by implicit differentiation.
x2y+y2= 6x
Solution:
1. Differentiate both sides of the equation with respect to xusing
implicit differentiation.
d
dx (x2y+y2) = d
dx (6x)
d
dx (x2y) + d
dx (y2)=6
2. Apply the product rule for differentiation of x2yand the chain
rule for differentiation of y2.
2xy +x2dy
dx + 2ydy
dx = 6
x2dy
dx + 2ydy
dx = 6 −2xy
dy
dx (x2+ 2y)=6−2xy
dy
dx =6−2xy
x2+ 2y
Therefore, dy
dx =6−2xy
x2+2y.
Question 10
Step-by-step solution:
Given equation: x2+y2= 9
1. Differentiate both sides of the equation with respect to x:
d
dx (x2) + d
dx (y2) = d
dx (9)
9
2. Applying the chain rule for d
dx (y2)gives us:
2x+ 2y·
dy
dx = 0
3. Now, solve for dy
dx :
2y·
dy
dx =−2x
dy
dx =
−2x
2y=
−x
y
Therefore, dy
dx =−x
y.Question 10: Find dy
dx for the equation x2+y2= 9
using implicit differentiation.
Step-by-step solution:
Given equation: x2+y2= 9
1. Differentiate both sides of the equation with respect to x:
d
dx (x2) + d
dx (y2) = d
dx (9)
2. Applying the chain rule for d
dx (y2)gives us:
2x+ 2y·
dy
dx = 0
3. Now, solve for dy
dx :
2y·
dy
dx =−2x
dy
dx =
−2x
2y=
−x
y
Therefore, dy
dx =−x
y.
Question 11
Find the derivative of the equation x2+y2= 9 with respect to x
using implicit differentiation.
Solution:
Given equation: x2+y2= 9
Step 1: Differentiate both sides of the equation with respect to x.
d
dx (x2) + d
dx (y2) = d
dx (9)
Step 2: Use the chain rule to differentiate y2with respect to x.
10
2ydy
dx = 0
Step 3: Solve for dy
dx .
dy
dx =0
2y= 0
Therefore, the derivative of the equation x2+y2= 9 with respect
to xis dy
dx = 0.Question 11:
Find the derivative of the equation x2+y2= 9 with respect to x
using implicit differentiation.
Solution:
Given equation: x2+y2= 9
Step 1: Differentiate both sides of the equation with respect to x.
d
dx (x2) + d
dx (y2) = d
dx (9)
Step 2: Use the chain rule to differentiate y2with respect to x.
2ydy
dx = 0
Step 3: Solve for dy
dx .
dy
dx =0
2y= 0
Therefore, the derivative of the equation x2+y2= 9 with respect
to xis dy
dx = 0.
Question 12
Find dy
dx by implicit differentiation: x2y+y2= 6
Solution: Step 1: Differentiate both sides of the equation with
respect to x.
d
dx (x2y+y2) = d
dx 6
2xy +x2dy
dx + 2ydy
dx = 0
Step 2: Now, solve for dy
dx .
11
2xy +x2dy
dx + 2ydy
dx = 0
2xy +x2dy
dx + 2ydy
dx = 0
x2dy
dx + 2ydy
dx =−2xy
dy
dx (x2+ 2y) = −2xy
dy
dx =
−2xy
x2+ 2y
Therefore, the derivative dy
dx of the given implicit equation is −2xy
x2+2y.Question
12:
Find dy
dx by implicit differentiation: x2y+y2= 6
Solution: Step 1: Differentiate both sides of the equation with
respect to x.
d
dx (x2y+y2) = d
dx 6
2xy +x2dy
dx + 2ydy
dx = 0
Step 2: Now, solve for dy
dx .
2xy +x2dy
dx + 2ydy
dx = 0
2xy +x2dy
dx + 2ydy
dx = 0
x2dy
dx + 2ydy
dx =−2xy
dy
dx (x2+ 2y) = −2xy
dy
dx =
−2xy
x2+ 2y
Therefore, the derivative dy
dx of the given implicit equation is −2xy
x2+2y.
Question 13
Question 13: Find dy
dx for the equation x2+y2= 25.
Solution:
Given equation is x2+y2= 25.
Taking the derivative of both sides with respect to x, we get:
12
d
dx (x2+y2) = d
dx (25)
Applying implicit differentiation gives:
d
dx (x2) + d
dx (y2)=0
2x+ 2ydy
dx = 0
Solving for dy
dx gives:
2ydy
dx =−2x
dy
dx =
−2x
2y
dy
dx =
−x
y
Therefore, the derivative dy
dx for the equation x2+y2= 25 is −x
y.Sure!
Here is a question and solution on Implicit Differentiation:
Question 13: Find dy
dx for the equation x2+y2= 25.
Solution:
Given equation is x2+y2= 25.
Taking the derivative of both sides with respect to x, we get:
d
dx (x2+y2) = d
dx (25)
Applying implicit differentiation gives:
d
dx (x2) + d
dx (y2)=0
2x+ 2ydy
dx = 0
Solving for dy
dx gives:
2ydy
dx =−2x
dy
dx =
−2x
2y
dy
dx =
−x
y
Therefore, the derivative dy
dx for the equation x2+y2= 25 is −x
y.
13
Question 14
Step-by-step solution: Given equation: x2+y2= 9
1. Differentiate both sides of the equation with respect to xusing
implicit differentiation:
d
dx [x2+y2] = d
dx 9
d
dx x2+d
dx y2= 0
2x+ 2ydy
dx = 0
2. Solve for dy
dx :
2ydy
dx =−2x
dy
dx =
−2x
2y
dy
dx =
−x
y
Therefore, dy
dx =
−x
y.Question 14: Find dy
dx for x2+y2= 9.
Step-by-step solution: Given equation: x2+y2= 9
1. Differentiate both sides of the equation with respect to xusing
implicit differentiation:
d
dx [x2+y2] = d
dx 9
d
dx x2+d
dx y2= 0
2x+ 2ydy
dx = 0
2. Solve for dy
dx :
2ydy
dx =−2x
dy
dx =
−2x
2y
dy
dx =
−x
y
Therefore, dy
dx =
−x
y.
14
Question 15
Step-by-step solution: 1. Start by differentiating both sides of the
equation with respect to x.
d
dx (x2+y2) = d
dx (25)
2x+ 2ydy
dx = 0
2. Now, isolate dy
dx on one side of the equation.
2ydy
dx =−2x
dy
dx =
−2x
2y
dy
dx =
−x
y
Therefore, dy
dx =−x
y.Question 15: Find dy
dx for the equation x2+y2=
25 using implicit differentiation.
Step-by-step solution: 1. Start by differentiating both sides of the
equation with respect to x.
d
dx (x2+y2) = d
dx (25)
2x+ 2ydy
dx = 0
2. Now, isolate dy
dx on one side of the equation.
2ydy
dx =−2x
dy
dx =
−2x
2y
dy
dx =
−x
y
Therefore, dy
dx =−x
y.
15
Question 2
Find dy
dx by implicit differentiation: x3+y3= 9xy.
Solution:
Given equation: x3+y3= 9xy
Taking derivative of both sides with respect to x:
d
dx (x3) + d
dx (y3) = d
dx (9xy)
3x2+ 3y2dy
dx = 9y+ 9xdy
dx
Moving terms involving dy
dx to one side:
3y2dy
dx
−9xdy
dx = 9y−3x2
Factor out dy
dx :
dy
dx (3y2
−9x)=9y−3x2
Now, solve for dy
dx :
dy
dx =9y−3x2
3y2−9x
Therefore, dy
dx =9y−3x2
3y2−9xQuestion 2:
Find dy
dx by implicit differentiation: x3+y3= 9xy.
Solution:
Given equation: x3+y3= 9xy
Taking derivative of both sides with respect to x:
d
dx (x3) + d
dx (y3) = d
dx (9xy)
3x2+ 3y2dy
dx = 9y+ 9xdy
dx
Moving terms involving dy
dx to one side:
3y2dy
dx
−9xdy
dx = 9y−3x2
Factor out dy
dx :
dy
dx (3y2
−9x)=9y−3x2
Now, solve for dy
dx :
2
dy
dx =9y−3x2
3y2−9x
Therefore, dy
dx =9y−3x2
3y2−9x
Question 3
Step-by-step Solution: Given equation: 2x2+ 3y2= 5xy
1. Differentiate both sides of the equation with respect to xusing
the rules of implicit differentiation.
d
dx (2x2) + d
dx (3y2) = d
dx (5xy)
2·2x+ 3 ·2ydy
dx = 5xdy
dx + 5y
2. Simplify the equation by isolating dy
dx on one side.
4x+ 6ydy
dx = 5xdy
dx + 5y
6ydy
dx
−5xdy
dx = 5y−4x
dy
dx (6y−5x)=5y−4x
dy
dx =5y−4x
6y−5x
Therefore, the derivative of ywith respect to xis 5y−4x
6y−5x.Question
3: Find dy
dx by implicit differentiation: 2x2+ 3y2= 5xy
Step-by-step Solution: Given equation: 2x2+ 3y2= 5xy
1. Differentiate both sides of the equation with respect to xusing
the rules of implicit differentiation.
d
dx (2x2) + d
dx (3y2) = d
dx (5xy)
2·2x+ 3 ·2ydy
dx = 5xdy
dx + 5y
2. Simplify the equation by isolating dy
dx on one side.
4x+ 6ydy
dx = 5xdy
dx + 5y
6ydy
dx
−5xdy
dx = 5y−4x
dy
dx (6y−5x)=5y−4x
dy
dx =5y−4x
6y−5x
3
Therefore, the derivative of ywith respect to xis 5y−4x
6y−5x.
Question 4
Step-by-step solution: Step 1: Differentiate both sides with re-
spect to x:
d
dx (x2) + d
dx (y2) = d
dx (4)
Step 2: Simplify using the chain rule for d
dx (y2)and the constant
rule for d
dx (4):
2x+ 2ydy
dx = 0
Step 3: Solve for dy
dx :
2ydy
dx =−2x
dy
dx =
−2x
2y=
−x
y
Therefore, the derivative of ywith respect to xis −x
y.Question 4:
Find dy
dx by implicit differentiation: x2+y2= 4.
Step-by-step solution: Step 1: Differentiate both sides with re-
spect to x:
d
dx (x2) + d
dx (y2) = d
dx (4)
Step 2: Simplify using the chain rule for d
dx (y2)and the constant
rule for d
dx (4):
2x+ 2ydy
dx = 0
Step 3: Solve for dy
dx :
2ydy
dx =−2x
dy
dx =
−2x
2y=
−x
y
Therefore, the derivative of ywith respect to xis −x
y.
Question 5
Find dy
dx by implicit differentiation:
3x2+ 4xy + 2y2= 12
Solution:
4
To find dy
dx by implicit differentiation, follow these steps:
Step 1: Differentiate both sides of the equation with respect to x:
d
dx (3x2) + d
dx (4xy) + d
dx (2y2) = d
dx (12)
Step 2: Simplify each term using the chain rule and product rule
as needed:
6x+ 4xdy
dx + 4y+ 4ydy
dx = 0
Step 3: Rearrange the terms to solve for dy
dx :
4xdy
dx + 4ydy
dx =−6x−4y
Step 4: Factor out dy
dx :
dy
dx (4x+ 4y) = −6x−4y
Step 5: Solve for dy
dx :
dy
dx =
−6x−4y
4x+ 4y
Question 5:
Find dy
dx by implicit differentiation:
3x2+ 4xy + 2y2= 12
Solution:
To find dy
dx by implicit differentiation, follow these steps:
Step 1: Differentiate both sides of the equation with respect to x:
d
dx (3x2) + d
dx (4xy) + d
dx (2y2) = d
dx (12)
Step 2: Simplify each term using the chain rule and product rule
as needed:
6x+ 4xdy
dx + 4y+ 4ydy
dx = 0
Step 3: Rearrange the terms to solve for dy
dx :
4xdy
dx + 4ydy
dx =−6x−4y
Step 4: Factor out dy
dx :
dy
dx (4x+ 4y) = −6x−4y
5
Step 5: Solve for dy
dx :
dy
dx =
−6x−4y
4x+ 4y
Question 6
Solution: Given equation: x2+y2= 1
Step 1: Differentiate both sides of the equation with respect to x
using implicit differentiation.
d
dx (x2) + d
dx (y2) = d
dx (1)
2x+ 2ydy
dx = 0
Step 2: Solve for dy
dx .
2x+ 2ydy
dx = 0
2ydy
dx =−2x
dy
dx =
−2x
2y
dy
dx =
−x
y
Therefore, dy
dx =−x
y.Question 6: Find dy
dx for x2+y2= 1 using implicit
differentiation.
Solution: Given equation: x2+y2= 1
Step 1: Differentiate both sides of the equation with respect to x
using implicit differentiation.
d
dx (x2) + d
dx (y2) = d
dx (1)
2x+ 2ydy
dx = 0
Step 2: Solve for dy
dx .
2x+ 2ydy
dx = 0
2ydy
dx =−2x
dy
dx =
−2x
2y
dy
dx =
−x
y
Therefore, dy
dx =−x
y.
6
Question 7
Step-by-step solution: To find dy
dx using implicit differentiation, we
follow these steps:
1. Differentiate both sides of the equation with respect to x. 2.
Apply the chain rule when differentiating terms involving y. 3. Solve
for dy
dx .
Given equation: x2+y2= 5
Step 1: Differentiate both sides with respect to x:
d
dx (x2) + d
dx (y2) = d
dx (5)
2x+ 2ydy
dx = 0
Step 2: Apply the chain rule to the term 2ydy
dx :
2x+ 2yd
dx (dy
dx )=0
2x+ 2yy′= 0
Step 3: Solve for dy
dx :
2yy′=−2x
y′=
−2x
2y
dy
dx =
−x
y
Therefore, the derivative of ywith respect to xis −x
y.Question 7:
Find dy
dx for the equation x2+y2= 5 using implicit differentiation.
Step-by-step solution: To find dy
dx using implicit differentiation, we
follow these steps:
1. Differentiate both sides of the equation with respect to x. 2.
Apply the chain rule when differentiating terms involving y. 3. Solve
for dy
dx .
Given equation: x2+y2= 5
Step 1: Differentiate both sides with respect to x:
d
dx (x2) + d
dx (y2) = d
dx (5)
2x+ 2ydy
dx = 0
Step 2: Apply the chain rule to the term 2ydy
dx :
2x+ 2yd
dx (dy
dx )=0
7
2x+ 2yy′= 0
Step 3: Solve for dy
dx :
2yy′=−2x
y′=
−2x
2y
dy
dx =
−x
y
Therefore, the derivative of ywith respect to xis −x
y.
Question 8
Step-by-step solution: 1. Differentiate both sides of the equation
with respect to x. 2. For x2, apply power rule to get 2x. 3. For
y2, differentiate implicitly using the chain rule to get 2ydy
dx . 4. The
derivative of a constant (9) with respect to xis 0. 5. Combine the
results to get 2x+ 2ydy
dx = 0. 6. Solve for dy
dx by isolating it: dy
dx =−x
y.
Therefore, dy
dx =−x
y.Question 8: Find dy
dx by implicit differentiation:
x2+y2= 9
Step-by-step solution: 1. Differentiate both sides of the equation
with respect to x. 2. For x2, apply power rule to get 2x. 3. For
y2, differentiate implicitly using the chain rule to get 2ydy
dx . 4. The
derivative of a constant (9) with respect to xis 0. 5. Combine the
results to get 2x+ 2ydy
dx = 0. 6. Solve for dy
dx by isolating it: dy
dx =−x
y.
Therefore, dy
dx =−x
y.
Question 9
Find dy
dx by implicit differentiation.
x2y+y2= 6x
Solution:
1. Differentiate both sides of the equation with respect to xusing
implicit differentiation.
d
dx (x2y+y2) = d
dx (6x)
d
dx (x2y) + d
dx (y2)=6
8
2. Apply the product rule for differentiation of x2yand the chain
rule for differentiation of y2.
2xy +x2dy
dx + 2ydy
dx = 6
x2dy
dx + 2ydy
dx = 6 −2xy
dy
dx (x2+ 2y)=6−2xy
dy
dx =6−2xy
x2+ 2y
Therefore, dy
dx =6−2xy
x2+2y.Question 9:
Find dy
dx by implicit differentiation.
x2y+y2= 6x
Solution:
1. Differentiate both sides of the equation with respect to xusing
implicit differentiation.
d
dx (x2y+y2) = d
dx (6x)
d
dx (x2y) + d
dx (y2)=6
2. Apply the product rule for differentiation of x2yand the chain
rule for differentiation of y2.
2xy +x2dy
dx + 2ydy
dx = 6
x2dy
dx + 2ydy
dx = 6 −2xy
dy
dx (x2+ 2y)=6−2xy
dy
dx =6−2xy
x2+ 2y
Therefore, dy
dx =6−2xy
x2+2y.
Question 10
Step-by-step solution:
Given equation: x2+y2= 9
1. Differentiate both sides of the equation with respect to x:
d
dx (x2) + d
dx (y2) = d
dx (9)
9
2. Applying the chain rule for d
dx (y2)gives us:
2x+ 2y·
dy
dx = 0
3. Now, solve for dy
dx :
2y·
dy
dx =−2x
dy
dx =
−2x
2y=
−x
y
Therefore, dy
dx =−x
y.Question 10: Find dy
dx for the equation x2+y2= 9
using implicit differentiation.
Step-by-step solution:
Given equation: x2+y2= 9
1. Differentiate both sides of the equation with respect to x:
d
dx (x2) + d
dx (y2) = d
dx (9)
2. Applying the chain rule for d
dx (y2)gives us:
2x+ 2y·
dy
dx = 0
3. Now, solve for dy
dx :
2y·
dy
dx =−2x
dy
dx =
−2x
2y=
−x
y
Therefore, dy
dx =−x
y.
Question 11
Find the derivative of the equation x2+y2= 9 with respect to x
using implicit differentiation.
Solution:
Given equation: x2+y2= 9
Step 1: Differentiate both sides of the equation with respect to x.
d
dx (x2) + d
dx (y2) = d
dx (9)
Step 2: Use the chain rule to differentiate y2with respect to x.
10
2ydy
dx = 0
Step 3: Solve for dy
dx .
dy
dx =0
2y= 0
Therefore, the derivative of the equation x2+y2= 9 with respect
to xis dy
dx = 0.Question 11:
Find the derivative of the equation x2+y2= 9 with respect to x
using implicit differentiation.
Solution:
Given equation: x2+y2= 9
Step 1: Differentiate both sides of the equation with respect to x.
d
dx (x2) + d
dx (y2) = d
dx (9)
Step 2: Use the chain rule to differentiate y2with respect to x.
2ydy
dx = 0
Step 3: Solve for dy
dx .
dy
dx =0
2y= 0
Therefore, the derivative of the equation x2+y2= 9 with respect
to xis dy
dx = 0.
Question 12
Find dy
dx by implicit differentiation: x2y+y2= 6
Solution: Step 1: Differentiate both sides of the equation with
respect to x.
d
dx (x2y+y2) = d
dx 6
2xy +x2dy
dx + 2ydy
dx = 0
Step 2: Now, solve for dy
dx .
11
2xy +x2dy
dx + 2ydy
dx = 0
2xy +x2dy
dx + 2ydy
dx = 0
x2dy
dx + 2ydy
dx =−2xy
dy
dx (x2+ 2y) = −2xy
dy
dx =
−2xy
x2+ 2y
Therefore, the derivative dy
dx of the given implicit equation is −2xy
x2+2y.Question
12:
Find dy
dx by implicit differentiation: x2y+y2= 6
Solution: Step 1: Differentiate both sides of the equation with
respect to x.
d
dx (x2y+y2) = d
dx 6
2xy +x2dy
dx + 2ydy
dx = 0
Step 2: Now, solve for dy
dx .
2xy +x2dy
dx + 2ydy
dx = 0
2xy +x2dy
dx + 2ydy
dx = 0
x2dy
dx + 2ydy
dx =−2xy
dy
dx (x2+ 2y) = −2xy
dy
dx =
−2xy
x2+ 2y
Therefore, the derivative dy
dx of the given implicit equation is −2xy
x2+2y.
Question 13
Question 13: Find dy
dx for the equation x2+y2= 25.
Solution:
Given equation is x2+y2= 25.
Taking the derivative of both sides with respect to x, we get:
12
d
dx (x2+y2) = d
dx (25)
Applying implicit differentiation gives:
d
dx (x2) + d
dx (y2)=0
2x+ 2ydy
dx = 0
Solving for dy
dx gives:
2ydy
dx =−2x
dy
dx =
−2x
2y
dy
dx =
−x
y
Therefore, the derivative dy
dx for the equation x2+y2= 25 is −x
y.Sure!
Here is a question and solution on Implicit Differentiation:
Question 13: Find dy
dx for the equation x2+y2= 25.
Solution:
Given equation is x2+y2= 25.
Taking the derivative of both sides with respect to x, we get:
d
dx (x2+y2) = d
dx (25)
Applying implicit differentiation gives:
d
dx (x2) + d
dx (y2)=0
2x+ 2ydy
dx = 0
Solving for dy
dx gives:
2ydy
dx =−2x
dy
dx =
−2x
2y
dy
dx =
−x
y
Therefore, the derivative dy
dx for the equation x2+y2= 25 is −x
y.
13
Question 14
Step-by-step solution: Given equation: x2+y2= 9
1. Differentiate both sides of the equation with respect to xusing
implicit differentiation:
d
dx [x2+y2] = d
dx 9
d
dx x2+d
dx y2= 0
2x+ 2ydy
dx = 0
2. Solve for dy
dx :
2ydy
dx =−2x
dy
dx =
−2x
2y
dy
dx =
−x
y
Therefore, dy
dx =
−x
y.Question 14: Find dy
dx for x2+y2= 9.
Step-by-step solution: Given equation: x2+y2= 9
1. Differentiate both sides of the equation with respect to xusing
implicit differentiation:
d
dx [x2+y2] = d
dx 9
d
dx x2+d
dx y2= 0
2x+ 2ydy
dx = 0
2. Solve for dy
dx :
2ydy
dx =−2x
dy
dx =
−2x
2y
dy
dx =
−x
y
Therefore, dy
dx =
−x
y.
14
Question 15
Step-by-step solution: 1. Start by differentiating both sides of the
equation with respect to x.
d
dx (x2+y2) = d
dx (25)
2x+ 2ydy
dx = 0
2. Now, isolate dy
dx on one side of the equation.
2ydy
dx =−2x
dy
dx =
−2x
2y
dy
dx =
−x
y
Therefore, dy
dx =−x
y.Question 15: Find dy
dx for the equation x2+y2=
25 using implicit differentiation.
Step-by-step solution: 1. Start by differentiating both sides of the
equation with respect to x.
d
dx (x2+y2) = d
dx (25)
2x+ 2ydy
dx = 0
2. Now, isolate dy
dx on one side of the equation.
2ydy
dx =−2x
dy
dx =
−2x
2y
dy
dx =
−x
y
Therefore, dy
dx =−x
y.
15
Question 2
Find dy
dx by implicit differentiation: x3+y3= 9xy.
Solution:
Given equation: x3+y3= 9xy
Taking derivative of both sides with respect to x:
d
dx (x3) + d
dx (y3) = d
dx (9xy)
3x2+ 3y2dy
dx = 9y+ 9xdy
dx
Moving terms involving dy
dx to one side:
3y2dy
dx
−9xdy
dx = 9y−3x2
Factor out dy
dx :
dy
dx (3y2
−9x)=9y−3x2
Now, solve for dy
dx :
dy
dx =9y−3x2
3y2−9x
Therefore, dy
dx =9y−3x2
3y2−9xQuestion 2:
Find dy
dx by implicit differentiation: x3+y3= 9xy.
Solution:
Given equation: x3+y3= 9xy
Taking derivative of both sides with respect to x:
d
dx (x3) + d
dx (y3) = d
dx (9xy)
3x2+ 3y2dy
dx = 9y+ 9xdy
dx
Moving terms involving dy
dx to one side:
3y2dy
dx
−9xdy
dx = 9y−3x2
Factor out dy
dx :
dy
dx (3y2
−9x)=9y−3x2
Now, solve for dy
dx :
2
dy
dx =9y−3x2
3y2−9x
Therefore, dy
dx =9y−3x2
3y2−9x
Question 3
Step-by-step Solution: Given equation: 2x2+ 3y2= 5xy
1. Differentiate both sides of the equation with respect to xusing
the rules of implicit differentiation.
d
dx (2x2) + d
dx (3y2) = d
dx (5xy)
2·2x+ 3 ·2ydy
dx = 5xdy
dx + 5y
2. Simplify the equation by isolating dy
dx on one side.
4x+ 6ydy
dx = 5xdy
dx + 5y
6ydy
dx
−5xdy
dx = 5y−4x
dy
dx (6y−5x)=5y−4x
dy
dx =5y−4x
6y−5x
Therefore, the derivative of ywith respect to xis 5y−4x
6y−5x.Question
3: Find dy
dx by implicit differentiation: 2x2+ 3y2= 5xy
Step-by-step Solution: Given equation: 2x2+ 3y2= 5xy
1. Differentiate both sides of the equation with respect to xusing
the rules of implicit differentiation.
d
dx (2x2) + d
dx (3y2) = d
dx (5xy)
2·2x+ 3 ·2ydy
dx = 5xdy
dx + 5y
2. Simplify the equation by isolating dy
dx on one side.
4x+ 6ydy
dx = 5xdy
dx + 5y
6ydy
dx
−5xdy
dx = 5y−4x
dy
dx (6y−5x)=5y−4x
dy
dx =5y−4x
6y−5x
3
Therefore, the derivative of ywith respect to xis 5y−4x
6y−5x.
Question 4
Step-by-step solution: Step 1: Differentiate both sides with re-
spect to x:
d
dx (x2) + d
dx (y2) = d
dx (4)
Step 2: Simplify using the chain rule for d
dx (y2)and the constant
rule for d
dx (4):
2x+ 2ydy
dx = 0
Step 3: Solve for dy
dx :
2ydy
dx =−2x
dy
dx =
−2x
2y=
−x
y
Therefore, the derivative of ywith respect to xis −x
y.Question 4:
Find dy
dx by implicit differentiation: x2+y2= 4.
Step-by-step solution: Step 1: Differentiate both sides with re-
spect to x:
d
dx (x2) + d
dx (y2) = d
dx (4)
Step 2: Simplify using the chain rule for d
dx (y2)and the constant
rule for d
dx (4):
2x+ 2ydy
dx = 0
Step 3: Solve for dy
dx :
2ydy
dx =−2x
dy
dx =
−2x
2y=
−x
y
Therefore, the derivative of ywith respect to xis −x
y.
Question 5
Find dy
dx by implicit differentiation:
3x2+ 4xy + 2y2= 12
Solution:
4
To find dy
dx by implicit differentiation, follow these steps:
Step 1: Differentiate both sides of the equation with respect to x:
d
dx (3x2) + d
dx (4xy) + d
dx (2y2) = d
dx (12)
Step 2: Simplify each term using the chain rule and product rule
as needed:
6x+ 4xdy
dx + 4y+ 4ydy
dx = 0
Step 3: Rearrange the terms to solve for dy
dx :
4xdy
dx + 4ydy
dx =−6x−4y
Step 4: Factor out dy
dx :
dy
dx (4x+ 4y) = −6x−4y
Step 5: Solve for dy
dx :
dy
dx =
−6x−4y
4x+ 4y
Question 5:
Find dy
dx by implicit differentiation:
3x2+ 4xy + 2y2= 12
Solution:
To find dy
dx by implicit differentiation, follow these steps:
Step 1: Differentiate both sides of the equation with respect to x:
d
dx (3x2) + d
dx (4xy) + d
dx (2y2) = d
dx (12)
Step 2: Simplify each term using the chain rule and product rule
as needed:
6x+ 4xdy
dx + 4y+ 4ydy
dx = 0
Step 3: Rearrange the terms to solve for dy
dx :
4xdy
dx + 4ydy
dx =−6x−4y
Step 4: Factor out dy
dx :
dy
dx (4x+ 4y) = −6x−4y
5
Step 5: Solve for dy
dx :
dy
dx =
−6x−4y
4x+ 4y
Question 6
Solution: Given equation: x2+y2= 1
Step 1: Differentiate both sides of the equation with respect to x
using implicit differentiation.
d
dx (x2) + d
dx (y2) = d
dx (1)
2x+ 2ydy
dx = 0
Step 2: Solve for dy
dx .
2x+ 2ydy
dx = 0
2ydy
dx =−2x
dy
dx =
−2x
2y
dy
dx =
−x
y
Therefore, dy
dx =−x
y.Question 6: Find dy
dx for x2+y2= 1 using implicit
differentiation.
Solution: Given equation: x2+y2= 1
Step 1: Differentiate both sides of the equation with respect to x
using implicit differentiation.
d
dx (x2) + d
dx (y2) = d
dx (1)
2x+ 2ydy
dx = 0
Step 2: Solve for dy
dx .
2x+ 2ydy
dx = 0
2ydy
dx =−2x
dy
dx =
−2x
2y
dy
dx =
−x
y
Therefore, dy
dx =−x
y.
6
Question 7
Step-by-step solution: To find dy
dx using implicit differentiation, we
follow these steps:
1. Differentiate both sides of the equation with respect to x. 2.
Apply the chain rule when differentiating terms involving y. 3. Solve
for dy
dx .
Given equation: x2+y2= 5
Step 1: Differentiate both sides with respect to x:
d
dx (x2) + d
dx (y2) = d
dx (5)
2x+ 2ydy
dx = 0
Step 2: Apply the chain rule to the term 2ydy
dx :
2x+ 2yd
dx (dy
dx )=0
2x+ 2yy′= 0
Step 3: Solve for dy
dx :
2yy′=−2x
y′=
−2x
2y
dy
dx =
−x
y
Therefore, the derivative of ywith respect to xis −x
y.Question 7:
Find dy
dx for the equation x2+y2= 5 using implicit differentiation.
Step-by-step solution: To find dy
dx using implicit differentiation, we
follow these steps:
1. Differentiate both sides of the equation with respect to x. 2.
Apply the chain rule when differentiating terms involving y. 3. Solve
for dy
dx .
Given equation: x2+y2= 5
Step 1: Differentiate both sides with respect to x:
d
dx (x2) + d
dx (y2) = d
dx (5)
2x+ 2ydy
dx = 0
Step 2: Apply the chain rule to the term 2ydy
dx :
2x+ 2yd
dx (dy
dx )=0
7
2x+ 2yy′= 0
Step 3: Solve for dy
dx :
2yy′=−2x
y′=
−2x
2y
dy
dx =
−x
y
Therefore, the derivative of ywith respect to xis −x
y.
Question 8
Step-by-step solution: 1. Differentiate both sides of the equation
with respect to x. 2. For x2, apply power rule to get 2x. 3. For
y2, differentiate implicitly using the chain rule to get 2ydy
dx . 4. The
derivative of a constant (9) with respect to xis 0. 5. Combine the
results to get 2x+ 2ydy
dx = 0. 6. Solve for dy
dx by isolating it: dy
dx =−x
y.
Therefore, dy
dx =−x
y.Question 8: Find dy
dx by implicit differentiation:
x2+y2= 9
Step-by-step solution: 1. Differentiate both sides of the equation
with respect to x. 2. For x2, apply power rule to get 2x. 3. For
y2, differentiate implicitly using the chain rule to get 2ydy
dx . 4. The
derivative of a constant (9) with respect to xis 0. 5. Combine the
results to get 2x+ 2ydy
dx = 0. 6. Solve for dy
dx by isolating it: dy
dx =−x
y.
Therefore, dy
dx =−x
y.
Question 9
Find dy
dx by implicit differentiation.
x2y+y2= 6x
Solution:
1. Differentiate both sides of the equation with respect to xusing
implicit differentiation.
d
dx (x2y+y2) = d
dx (6x)
d
dx (x2y) + d
dx (y2)=6
8
2. Apply the product rule for differentiation of x2yand the chain
rule for differentiation of y2.
2xy +x2dy
dx + 2ydy
dx = 6
x2dy
dx + 2ydy
dx = 6 −2xy
dy
dx (x2+ 2y)=6−2xy
dy
dx =6−2xy
x2+ 2y
Therefore, dy
dx =6−2xy
x2+2y.Question 9:
Find dy
dx by implicit differentiation.
x2y+y2= 6x
Solution:
1. Differentiate both sides of the equation with respect to xusing
implicit differentiation.
d
dx (x2y+y2) = d
dx (6x)
d
dx (x2y) + d
dx (y2)=6
2. Apply the product rule for differentiation of x2yand the chain
rule for differentiation of y2.
2xy +x2dy
dx + 2ydy
dx = 6
x2dy
dx + 2ydy
dx = 6 −2xy
dy
dx (x2+ 2y)=6−2xy
dy
dx =6−2xy
x2+ 2y
Therefore, dy
dx =6−2xy
x2+2y.
Question 10
Step-by-step solution:
Given equation: x2+y2= 9
1. Differentiate both sides of the equation with respect to x:
d
dx (x2) + d
dx (y2) = d
dx (9)
9
2. Applying the chain rule for d
dx (y2)gives us:
2x+ 2y·
dy
dx = 0
3. Now, solve for dy
dx :
2y·
dy
dx =−2x
dy
dx =
−2x
2y=
−x
y
Therefore, dy
dx =−x
y.Question 10: Find dy
dx for the equation x2+y2= 9
using implicit differentiation.
Step-by-step solution:
Given equation: x2+y2= 9
1. Differentiate both sides of the equation with respect to x:
d
dx (x2) + d
dx (y2) = d
dx (9)
2. Applying the chain rule for d
dx (y2)gives us:
2x+ 2y·
dy
dx = 0
3. Now, solve for dy
dx :
2y·
dy
dx =−2x
dy
dx =
−2x
2y=
−x
y
Therefore, dy
dx =−x
y.
Question 11
Find the derivative of the equation x2+y2= 9 with respect to x
using implicit differentiation.
Solution:
Given equation: x2+y2= 9
Step 1: Differentiate both sides of the equation with respect to x.
d
dx (x2) + d
dx (y2) = d
dx (9)
Step 2: Use the chain rule to differentiate y2with respect to x.
10
2ydy
dx = 0
Step 3: Solve for dy
dx .
dy
dx =0
2y= 0
Therefore, the derivative of the equation x2+y2= 9 with respect
to xis dy
dx = 0.Question 11:
Find the derivative of the equation x2+y2= 9 with respect to x
using implicit differentiation.
Solution:
Given equation: x2+y2= 9
Step 1: Differentiate both sides of the equation with respect to x.
d
dx (x2) + d
dx (y2) = d
dx (9)
Step 2: Use the chain rule to differentiate y2with respect to x.
2ydy
dx = 0
Step 3: Solve for dy
dx .
dy
dx =0
2y= 0
Therefore, the derivative of the equation x2+y2= 9 with respect
to xis dy
dx = 0.
Question 12
Find dy
dx by implicit differentiation: x2y+y2= 6
Solution: Step 1: Differentiate both sides of the equation with
respect to x.
d
dx (x2y+y2) = d
dx 6
2xy +x2dy
dx + 2ydy
dx = 0
Step 2: Now, solve for dy
dx .
11
2xy +x2dy
dx + 2ydy
dx = 0
2xy +x2dy
dx + 2ydy
dx = 0
x2dy
dx + 2ydy
dx =−2xy
dy
dx (x2+ 2y) = −2xy
dy
dx =
−2xy
x2+ 2y
Therefore, the derivative dy
dx of the given implicit equation is −2xy
x2+2y.Question
12:
Find dy
dx by implicit differentiation: x2y+y2= 6
Solution: Step 1: Differentiate both sides of the equation with
respect to x.
d
dx (x2y+y2) = d
dx 6
2xy +x2dy
dx + 2ydy
dx = 0
Step 2: Now, solve for dy
dx .
2xy +x2dy
dx + 2ydy
dx = 0
2xy +x2dy
dx + 2ydy
dx = 0
x2dy
dx + 2ydy
dx =−2xy
dy
dx (x2+ 2y) = −2xy
dy
dx =
−2xy
x2+ 2y
Therefore, the derivative dy
dx of the given implicit equation is −2xy
x2+2y.
Question 13
Question 13: Find dy
dx for the equation x2+y2= 25.
Solution:
Given equation is x2+y2= 25.
Taking the derivative of both sides with respect to x, we get:
12
d
dx (x2+y2) = d
dx (25)
Applying implicit differentiation gives:
d
dx (x2) + d
dx (y2)=0
2x+ 2ydy
dx = 0
Solving for dy
dx gives:
2ydy
dx =−2x
dy
dx =
−2x
2y
dy
dx =
−x
y
Therefore, the derivative dy
dx for the equation x2+y2= 25 is −x
y.Sure!
Here is a question and solution on Implicit Differentiation:
Question 13: Find dy
dx for the equation x2+y2= 25.
Solution:
Given equation is x2+y2= 25.
Taking the derivative of both sides with respect to x, we get:
d
dx (x2+y2) = d
dx (25)
Applying implicit differentiation gives:
d
dx (x2) + d
dx (y2)=0
2x+ 2ydy
dx = 0
Solving for dy
dx gives:
2ydy
dx =−2x
dy
dx =
−2x
2y
dy
dx =
−x
y
Therefore, the derivative dy
dx for the equation x2+y2= 25 is −x
y.
13
Question 14
Step-by-step solution: Given equation: x2+y2= 9
1. Differentiate both sides of the equation with respect to xusing
implicit differentiation:
d
dx [x2+y2] = d
dx 9
d
dx x2+d
dx y2= 0
2x+ 2ydy
dx = 0
2. Solve for dy
dx :
2ydy
dx =−2x
dy
dx =
−2x
2y
dy
dx =
−x
y
Therefore, dy
dx =
−x
y.Question 14: Find dy
dx for x2+y2= 9.
Step-by-step solution: Given equation: x2+y2= 9
1. Differentiate both sides of the equation with respect to xusing
implicit differentiation:
d
dx [x2+y2] = d
dx 9
d
dx x2+d
dx y2= 0
2x+ 2ydy
dx = 0
2. Solve for dy
dx :
2ydy
dx =−2x
dy
dx =
−2x
2y
dy
dx =
−x
y
Therefore, dy
dx =
−x
y.
14
Question 15
Step-by-step solution: 1. Start by differentiating both sides of the
equation with respect to x.
d
dx (x2+y2) = d
dx (25)
2x+ 2ydy
dx = 0
2. Now, isolate dy
dx on one side of the equation.
2ydy
dx =−2x
dy
dx =
−2x
2y
dy
dx =
−x
y
Therefore, dy
dx =−x
y.Question 15: Find dy
dx for the equation x2+y2=
25 using implicit differentiation.
Step-by-step solution: 1. Start by differentiating both sides of the
equation with respect to x.
d
dx (x2+y2) = d
dx (25)
2x+ 2ydy
dx = 0
2. Now, isolate dy
dx on one side of the equation.
2ydy
dx =−2x
dy
dx =
−2x
2y
dy
dx =
−x
y
Therefore, dy
dx =−x
y.
15
