Master Implicit Differentiation: Enhance Your Calculus III Skills

Question 1

Find the derivative of the implicit function defined by the equation cos(x*y) = x^2 + y^2.

Accepted Answer

dy/dx = (y*sin(x*y) - 2x)/(x*sin(x*y) - 2y)

Answer

dy/dx = (y*sin(x*y) + 2x)/(x*sin(x*y) - 2y)

Answer

dy/dx = (y*sin(x*y) - 2x)/(x*sin(x*y) + 2y)

Answer

dy/dx = (y*sin(x*y) + 2x)/(x*sin(x*y) + 2y)

Question 2

Which step is not involved in the process of implicit differentiation?

Accepted Answer

Solving the equation for y explicitly

Answer

Differentiating both sides of the equation with respect to x

Answer

Substituting the derivative of y into the equation

Answer

Simplifying the resulting equation

Question 3

Find the derivative of the function defined implicitly by the equation sin(x + y) = y.

Accepted Answer

y' = cos(x + y) / (1 + sin(x + y))

Answer

y' = cos(x + y) / (1 - sin(x + y))

Answer

y' = sin(x + y) / (1 + cos(x + y))

Answer

y' = sin(x + y) / (1 - cos(x + y))

Question 4

Find the derivative of the function defined implicitly by the equation ln(x^2 + y^2) = y.

Accepted Answer

y' = (x) / (x^2 + y^2 - y)

Answer

y' = (y) / (x^2 + y^2 - x)

Answer

y' = (x) / (x^2 + y^2 + y)

Answer

y' = (y) / (x^2 + y^2 + x)

Question 5

Consider the implicit function: e^(x+y) = xy + 1. Given y = 1 at x = 0, calculate y'(0).

Accepted Answer

y'(0) = 1

Answer

y'(0) = 0

Answer

y'(0) = 2

Answer

y'(0) = -1

Question 6

Implicit differentiation fails to find a meaningful derivative for a function defined by F(x, y) = 0 when:

Accepted Answer

The derivative of F(x, y) with respect to x does not exist.

Answer

The function is defined explicitly.

Answer

The function is a constant.

Answer

The function is nonlinear.

Question 7

Describe the geometric interpretation of implicit differentiation.

Accepted Answer

It calculates the slope of the tangent line to the level curve of the implicit function.

Answer

It calculates the area under the curve of the implicit function.

Answer

It calculates the volume of the solid generated by rotating the graph of the implicit function.

Answer

It calculates the integral of the implicit function.

Question 8

Which of the following is a limitation of implicit differentiation?

Accepted Answer

It can be challenging to apply to implicit functions that are highly complex or involve multiple variables.

Answer

It cannot be used to find the derivative of any implicit function.

Answer

It can only be used to find the first derivative of an implicit function.

Answer

It can only be used to find the derivative of an implicit function that is defined by a polynomial equation.

Question 9

Find the derivative of the implicit function defined by the equation y^3 + x^2y = 1 using implicit differentiation.

Accepted Answer

dy/dx = (2xy - y^2) / (x^2 + 3y^2)

Answer

dy/dx = (x^2 - y^2) / (x^2 + 3y^2)

Answer

dy/dx = (2xy + y^2) / (x^2 + 3y^2)

Answer

dy/dx = (x^2 + y^2) / (2xy - y^2)

Question 10

Find the derivative of the implicit function defined by the equation ln(x+2y) - sin(x-y) = 0 using implicit differentiation.

Accepted Answer

dy/dx = (cos(x-y) / (1+2y)) / (1 / (x+2y))

Answer

dy/dx = (sin(x-y) / (1+2y)) / (1 / (x+2y))

Answer

dy/dx = (sin(x-y) / (1+2y)) / (1 / (x+2y))

Question 11

Which of the following is NOT a benefit of using implicit differentiation?

Accepted Answer

Can be used to find derivatives of functions that are not continuous at certain points.

Answer

Can be used to find derivatives of functions that are not explicitly defined.

Answer

Can be used to find derivatives of functions that are not differentiable at certain points.

Answer

Can be used to find derivatives of functions that are defined by more than one equation.

Question 12

Which of the following techniques can be used to find the derivative of an implicitly defined function?

Accepted Answer

Implicit differentiation

Answer

Chain rule

Answer

Product rule

Answer

Quotient rule

Question 13

Which of the following is NOT a benefit of using implicit differentiation?

Accepted Answer

It can be used to find derivatives of functions that involve inequalities.

Answer

It can be used to find derivatives of functions defined implicitly.

Answer

It allows for the differentiation of continuously differentiable functions.

Answer

It provides a systematic method for finding derivatives of implicitly defined functions.

Question 14

Implicit differentiation is most commonly used for which of the following?

Accepted Answer

Finding derivatives of functions that are implicitly defined

Answer

Finding derivatives of functions defined explicitly

Answer

Finding integrals of implicitly defined functions

Answer

Solving differential equations

Question 15

Which of the following represents the implicit function for a circle with radius 'r' centered at the origin?

Accepted Answer

x^2 + y^2 = r^2

Answer

(x + y)^2 = r^2

Answer

x^2 - y^2 = r^2

Question 16

Which of the following is NOT a valid step in the process of implicit differentiation?

Accepted Answer

Solve for y before differentiating.

Answer

Differentiate both sides of the equation with respect to x.

Answer

Treat y as a function of x and use the chain rule.

Question 17

Find the derivative of y with respect to x for the implicit function: x² + y² = 25

Accepted Answer

-x/y

Answer

x/y

Answer

y/x

Question 18

What is the slope of the tangent line to the curve defined by the equation x³ + y³ = 6xy at the point (2, 2)?

Accepted Answer

-1

Answer

2

Answer

1

Answer

0

Question 19

Find dy/dx for the implicit function: sin(x + y) = x²y

Accepted Answer

(2xy - cos(x + y)) / (cos(x + y) - x²)

Answer

(2xy - cos(x + y)) / (cos(x + y) + x²)

Answer

(2xy + cos(x + y)) / (cos(x + y) - x²)

Answer

(2xy + cos(x + y)) / (cos(x + y) + x²)

Question 20

Given the equation x² + xy + y² = 7, find the equation of the tangent line at the point (1, 2).

Accepted Answer

y = -4x + 6

Answer

y = 4x - 2

Answer

y = -2x + 4

Question 21

The equation x² + y² = 1 represents a circle. What is the derivative of y with respect to x at the point (√2/2, √2/2)?

Accepted Answer

-1

Answer

1

Answer

√2

Answer

0

Question 22

Find the second derivative (d²y/dx²) of the implicit function: x² + y² = 1

Accepted Answer

-1/y³

Answer

-1/x³

Answer

1/y³

Answer

1/x³

Question 23

What is the derivative of y with respect to x for the equation e^(xy) = x + y?

Accepted Answer

(1 - ye^(xy)) / (xe^(xy) - 1)

Answer

(1 + ye^(xy)) / (xe^(xy) + 1)

Answer

(1 - ye^(xy)) / (xe^(xy) + 1)

Question 24

Given the equation ln(x + y) = x²y, find dy/dx.

Accepted Answer

(2xy(x + y) - 1) / (1 - x² (x + y))

Answer

(2xy(x + y) + 1) / (1 - x² (x + y))

Answer

(2xy(x + y) - 1) / (1 + x² (x + y))

Answer

(2xy(x + y) + 1) / (1 + x² (x + y))

Question 25

Which of the following steps is NOT involved in the process of implicit differentiation?

Accepted Answer

Isolating y in terms of x

Answer

Treating y as a function of x

Answer

Differentiating both sides of the equation with respect to x

Answer

Using the chain rule where necessary

Question 26

The equation x² - y² = 1 defines a hyperbola. What is the derivative of y with respect to x at the point (2, √3)?

Accepted Answer

2/√3

Answer

-√3/2

Answer

√3/2

Answer

-2/√3

Question 27

Which of the following is the derivative of the implicit function x^2 + y^2 = 25?

Accepted Answer

x/y

Answer

x - y

Answer

2x

Answer

y/x

Question 28

Find the derivative of the implicit function defined by the equation xy - sin(y) = x^2 + 1.

Accepted Answer

(x+ycos(y)) / (x - ysin(y))

Answer

(x-ycos(y)) / (x + ysin(y))

Answer

(x-yscos(y)) / (x + ysin(y))

Answer

(y+xsin(y)) / (x - ycos(y))

Question 29

Which of the following techniques can be used to find the derivative of the implicit function x^3 + y^3 = 9?

Accepted Answer

Cube the derivative of both sides.

Answer

Separation of variables.

Answer

Logarithmic differentiation.

Answer

Implicit differentiation using the chain rule.

Question 30

Find the slope of the tangent line to the curve defined by the implicit function 2x^2 + 3xy - 5y^2 = 0 at the point (2, 1).

Accepted Answer

2/5

Answer

-5/2

Answer

5/2

Answer

-2/5

Question 31

Which of the following is a valid step in the implicit differentiation of the equation y^2 = 4x?

Accepted Answer

Differentiate both sides with respect to x.

Answer

Substitute y = 2sqrt(x).

Answer

Solve for y first.

Answer

Apply the chain rule to the left side only.

Question 32

Consider the implicit function x^2 + e^(xy) = y. Find dy/dx.

Accepted Answer

-(x + ye^(xy)) / (2x + ye^(xy) + y^2xe^(xy))

Answer

(x + ye^(xy)) / (2x + ye^(xy) + y^2xe^(xy))

Answer

-(2x + ye^(xy)) / (x + ye^(xy) + y^2xe^(xy))

Answer

(2x + ye^(xy)) / (x + ye^(xy) + y^2xe^(xy))

Question 33

Find the derivative of the implicit function defined by the equation sin(x + y) = y^2.

Accepted Answer

(cos(x + y) - 2yy') / (1 + 2y')

Answer

(cos(x + y) + 2yy') / (1 - 2y')

Answer

(cos(x + y) - 2yy') / (1 - 2y')

Answer

(cos(x + y) + 2yy') / (1 + 2y')

Question 34

Which of the following is the derivative of the implicit function x^4 - y^2 = 1?

Accepted Answer

(2x^3) / y

Answer

(-2x^3) / y

Answer

-y / (2x^3)

Answer

y / (2x^3)

Question 35

Consider the implicit function y = x^2 + sin(y). If x = 1, find dy/dx.

Accepted Answer

3

Answer

-2

Answer

2

Answer

-3

Question 36

Which of the following is a limitation of implicit differentiation?

Accepted Answer

It cannot be used to find the derivative of an explicit function.

Answer

It requires logarithmic differentiation for all functions.

Answer

It is only applicable to functions defined in terms of x and y.