Master Implicit Differentiation: A Comprehensive Guide for Calculus I Students

Question 1

Which of the following techniques can be used to find the derivative of the implicitly defined function sin(xy) = y^2 ?

Accepted Answer

Chain rule

Answer

Product rule

Answer

Quotient rule

Answer

Logarithmic differentiation

Question 2

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

Accepted Answer

y' = (x^2y - e^(xy) - 1) / (xe^(xy) + y/x)

Answer

y' = (x^2y + e^(xy) - 1) / (xe^(xy) + y/x)

Answer

y' = (x^2y - e^(xy) + 1) / (xe^(xy) + y/x)

Answer

y' = (x^2y + e^(xy) + 1) / (xe^(xy) + y/x)

Question 3

Which of the following equations is the implicit derivative of the function defined by x^3 + y^3 = 9xy ?

Accepted Answer

x^2 - y^2 = 9x - 9y

Answer

x^2 + y^2 = 9x + 9y

Answer

x^2 - y^2 = 9x + 9y

Answer

x^2 + y^2 = 9x - 9y

Question 4

Which of the following equations represents an implicit function?

Accepted Answer

x^2 + y^2 = 25

Answer

y = x^2 + 1

Answer

x + y = 5

Answer

y = sin(x)

Question 5

What is the partial derivative of y with respect to x (∂y/∂x) for the implicit function x² + y² - 4 = 0?

Accepted Answer

-x/y

Answer

y/x

Answer

x² - y²

Answer

4

Question 6

How do you use implicit differentiation to find critical points of an implicitly defined function?

Accepted Answer

Find the points where the first derivative is zero or undefined.

Answer

Find the points where the function is differentiable.

Answer

Find the points where the second derivative is zero.

Answer

Find the points where the function is continuous.

Question 7

Which of the following formulas is used for implicit differentiation?

Accepted Answer

dy/dx = fy/fx

Answer

dy/dx = -fx/fy

Answer

dy/dx = fx/fy

Answer

dy/dx = -fy/fx

Question 8

Calculate the derivative of the implicit function defined by x^3 + y^3 = 2xy using implicit differentiation.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 9

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

Accepted Answer

Substitute y for dy/dx.

Answer

Differentiate both sides of the equation with respect to x.

Answer

Simplify the resulting equation.

Answer

Solve for dy/dx.

Question 10

Which of the following techniques can be used to find the derivative of an implicit function when the function is not readily solvable for y?

Accepted Answer

Logarithmic differentiation

Answer

Product rule

Answer

Chain rule

Answer

Quotient rule

Question 11

Which of the following is a limitation of implicit differentiation?

Accepted Answer

It cannot be used to find the derivative of a function that is not defined implicitly.

Answer

It can only be used to find the first derivative.

Answer

It only works for functions that are linear.

Answer

It is more difficult than explicit differentiation.

Question 12

Which of the following applications of implicit differentiation is used to find the slope of a tangent line to a curve?

Accepted Answer

Finding the derivative of an implicit function.

Answer

Finding the area under a curve.

Answer

Solving differential equations.

Answer

Finding the volume of a solid of revolution.

Question 13

Which of the following functions CANNOT be differentiated implicitly?

Accepted Answer

$x^2 + |y| = 1$

Answer

$x^2 + y^2 = 1$

Answer

$x^3 - y^3 = 1$

Answer

$e^x + ln(y) = 1$

Question 14

Find $\frac{dy}{dx}$ for the equation $xy + sin(y) = 1$.

Accepted Answer

$\frac{dy}{dx} = \frac{-y}{x + cos(y)}$

Answer

$\frac{dy}{dx} = \frac{y}{x + cos(y)}$

Answer

$\frac{dy}{dx} = \frac{1}{x + cos(y)}$

Answer

$\frac{dy}{dx} = \frac{-y}{cos(y)}$

Question 15

Find $\frac{dy}{dx}$ for the equation $ln(y) = x^2 + y^2$.

Accepted Answer

$\frac{dy}{dx} = \frac{2xy}{1-2y^2}$

Answer

$\frac{dy}{dx} = \frac{2x}{1-2y}$

Answer

$\frac{dy}{dx} = \frac{2xy}{1+2y^2}$

Answer

$\frac{dy}{dx} = \frac{2x}{1+2y}$

Question 16

The equation $sin(x+y) = y^2 cos(x)$ defines $y$ implicitly as a function of $x$. Find $\frac{dy}{dx}$.

Accepted Answer

$\frac{dy}{dx} = \frac{cos(x+y) - y^2 sin(x)}{2y cos(x) - cos(x+y)}$

Answer

$\frac{dy}{dx} = \frac{cos(x+y)}{2y cos(x)}$

Answer

$\frac{dy}{dx} = \frac{-y^2 sin(x)}{2y cos(x)}$

Question 17

Given the equation $y = x^y$, find $\frac{dy}{dx}$.

Accepted Answer

$\frac{dy}{dx} = \frac{y x^{y-1}}{1 - x^y ln(x)}$

Answer

$\frac{dy}{dx} = \frac{y x^{y-1}}{1 + x^y ln(x)}$

Answer

$\frac{dy}{dx} = \frac{x^y}{1 - x^y ln(x)}$

Question 18

What is the slope of the tangent line to the curve defined by the equation  x³ + y³ = 6xy at the point (3, 3)?

Accepted Answer

-1

Answer

0

Answer

1

Answer

undefined

Question 19

Find the second derivative, d²y/dx², for the equation x² + y² = 1.

Accepted Answer

-1/y³

Answer

1/y³

Answer

1/x³

Answer

-1/x³

Question 20

Which of the following equations represents a possible solution to the differential equation dy/dx = -x/y?

Accepted Answer

x² + y² = 4

Answer

x² - y² = 4

Answer

y = x² + 4

Answer

y = -x² + 4

Question 21

When finding the derivative of y with respect to x using implicit differentiation, which of the following is NOT a necessary step?

Accepted Answer

Solve explicitly for y in terms of x before differentiating.

Answer

Differentiate both sides of the equation with respect to x.

Answer

Treat y as a function of x, using the chain rule when differentiating terms involving y.

Question 22

What is the equation of the tangent line to the curve defined by x²y + y² = 1 at the point (0, 1)?

Accepted Answer

y = 1

Answer

y = -x + 1

Answer

x = 0

Answer

y = x + 1

Question 23

Find the derivative, dy/dx, of the equation x² + y² = 25.

Accepted Answer

dy/dx = -x/y

Answer

dy/dx = -y/x

Answer

dy/dx = x/y

Answer

dy/dx = 2x/2y

Question 24

What is the correct result of implicitly differentiating the equation xy + sin(y) = 1 with respect to x?

Accepted Answer

y + x(dy/dx) + cos(y)(dy/dx) = 0

Answer

x + y(dy/dx) + cos(y) = 0

Answer

1 + (dy/dx) + cos(y)(dy/dx) = 0

Answer

y + x(dy/dx) - sin(y)(dy/dx) = 0

Question 25

If y is defined implicitly by the equation e^y = xy² + 1, find dy/dx in terms of x and y.

Accepted Answer

dy/dx = y² / (e^y - 2xy)

Answer

dy/dx = y² / (e^y + 2xy)

Answer

dy/dx = 2xy / (e^y - y²)

Answer

dy/dx = e^y / (y² - 2xy)

Question 26

Find the second derivative, d²y/dx², for the relation x²y + y³ = 10 in terms of x, y, and dy/dx.

Accepted Answer

d²y/dx² = (-2y - 4x(dy/dx) - 6y(dy/dx)²) / (x² + 3y²)

Answer

d²y/dx² = (-2y - 2x(dy/dx)) / (x² + 3y²)

Answer

d²y/dx² = (-2 - 4x(dy/dx)) / (x² + 3y²)

Question 27

The equation sin(x + y) = y²cos(x) defines y implicitly as a function of x. What is the value of dy/dx at the point (0, 0)?

Accepted Answer

1

Answer

0

Answer

-1

Question 28

Which of the following is the primary advantage of implicit differentiation?

Accepted Answer

Implicit differentiation allows us to find the derivative of a function even when it's difficult or impossible to solve explicitly for y in terms of x.

Answer

Implicit differentiation is necessary when the function is defined by a transcendental equation.

Answer

Implicit differentiation is the only way to find the derivative of a function involving both x and y.

Answer

Implicit differentiation simplifies the process of finding derivatives for all types of functions.

Question 29

The curve is defined implicitly by the equation $(x^2 + y^2)^2 = 4x^2y$.  Find the slope of the tangent line to the curve at the point $(1, 1)$.

Accepted Answer

0

Answer

1

Answer

-1

Answer

2

Question 30

Given the equation $x^2 + y^2 = 25$, find the value of $\frac{d^2y}{dx^2}$ at the point $(3, 4)$.

Accepted Answer

-$\frac{25}{64}$

Answer

$\frac{25}{64}$

Answer

-$\frac{7}{8}$

Answer

-$\frac{9}{16}$

Question 31

Find $\frac{dy}{dx}$ for the equation $x^2 + y^2 = 25$.

Accepted Answer

$\frac{dy}{dx} = -\frac{x}{y}$

Answer

$\frac{dy}{dx} = \frac{x}{y}$

Answer

$\frac{dy}{dx} = \frac{y}{x}$

Answer

$\frac{dy}{dx} = \frac{2x}{2y}$

Question 32

What is the slope of the tangent line to the curve $x^3 + y^3 - 6xy = 0$ at the point $(3,3)$?

Accepted Answer

-1

Answer

0

Answer

3

Answer

1

Question 33

If $y$ is defined implicitly by the equation $sin(xy) = y$, find $\frac{dy}{dx}$.

Accepted Answer

$\frac{dy}{dx} = \frac{y\cos(xy)}{1 - x\cos(xy)}$

Answer

$\frac{dy}{dx} = \frac{y}{1 - x\cos(xy)}$

Answer

$\frac{dy}{dx} = \frac{\cos(xy)}{1 - \cos(xy)}$

Answer

$\frac{dy}{dx} = \frac{y\cos(xy)}{x\cos(xy) - 1}$

Question 34

Which of the following correctly represents the first step in finding $\frac{dy}{dx}$ for the equation $e^{xy} = 2x + y$?

Accepted Answer

$e^{xy}(y + x\frac{dy}{dx}) = 2 + \frac{dy}{dx}$

Answer

$e^{xy}\frac{dy}{dx} = 2 + \frac{dy}{dx}$

Answer

$e^{xy}\frac{dy}{dx} = 2$

Answer

$e^{xy}(x + y) = 2 + \frac{dy}{dx}$

Question 35

At what points on the curve $2y^2 = x^3 + x^2$ is the tangent line horizontal?

Accepted Answer

(0, 0) and (-$\frac{2}{3}$, $\pm \frac{\sqrt{6}}{9}$)

Answer

(-1, 0) and (1, 0)

Answer

(0, 0) only

Answer

($\frac{2}{3}$, $\pm \frac{\sqrt{6}}{9}$)

Question 36

If $x^2 + xy + y^2 = 7$, what is the value of $\frac{dy}{dx}$ at the point where $x = 1$ and $y = 2$?

Accepted Answer

-$\frac{4}{5}$

Answer

$\frac{3}{5}$

Answer

-1

Answer

2

Question 37

Find the second derivative of the implicit function: x^3 + y^3 = 6xy

Accepted Answer

y'' = (2x^2y - 6xy^2) / (y^3 - 3x^2y)

Answer

y'' = (2xy^2 - 6x^2y) / (y^3 - 3x^2y)

Answer

y'' = (6xy - 3x^2y) / (y^3 - 3x^2y)

Answer

y'' = (6xy^2 - 2x^2y) / (y^3 - 3x^2y)

Question 38

Which of the following equations implicitly defines y as a function of x? I. x^2 - y^3 = 4 II. y / x = 5 III. sin(x + y) = 0

Accepted Answer

I and III

Answer

None of the above

Answer

I and II

Answer

II and III

Question 39

Given the implicit function xy - x^2y = 1, find dy/dx at the point (1, 2)

Accepted Answer

1/3

Answer

-1/3

Answer

1/2

Question 40

Given the implicit function x^2 + y^2 - 4 = 0, find the value of d^2y/dx^2 at the point (2, 0)

Accepted Answer

-1/2

Answer

1/2

Answer

2

Answer

-2

Question 41

Find dy/dx for the equation sin(x + y) = y.

Accepted Answer

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

Answer

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

Answer

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

Question 42

Which of the following equations represents the derivative dy/dx obtained from the equation x² + y² = 1?

Accepted Answer

dy/dx = -x/y

Answer

dy/dx = -y/x

Answer

dy/dx = y/x

Answer

dy/dx = x/y

Question 43

What is the slope of the tangent line to the curve xy² = 1 at the point (1,1)?

Accepted Answer

-2

Answer

-1

Answer

2

Question 44

Find the derivative of y with respect to x for the equation x² + y² = 25.

Accepted Answer

dy/dx = -x/y

Answer

dy/dx = x/y

Answer

dy/dx = y/x

Answer

dy/dx = -y/x

Question 45

Find the slope of the tangent line to the curve x³ + y³ = 6xy at the point (3, 3).

Accepted Answer

1

Answer

0

Answer

Undefined

Answer

-1

Question 46

Given the implicit function xy = 4, find d²y/dx².

Accepted Answer

d²y/dx² = 8/x³

Answer

d²y/dx² = 4/x²

Answer

d²y/dx² = -8/x³

Answer

d²y/dx² = -4/x²

Question 47

Find the equation of the tangent line to the curve y² - x² = 16 at the point (3, 5).

Accepted Answer

y = (3/5)x + 16/5

Answer

y = -(3/5)x + 16/5

Answer

y = -(5/3)x + 4/3

Answer

y = (5/3)x - 4/3

Question 48

If sin(xy) = x, find dy/dx.

Accepted Answer

dy/dx = (1 - y*cos(xy))/(x*cos(xy))

Answer

dy/dx = (1 - x*cos(xy))/(y*cos(xy))

Answer

dy/dx = (1 + y*cos(xy))/(x*cos(xy))

Question 49

Find the derivative of y with respect to x for the equation e^(xy) = x + y.

Accepted Answer

dy/dx = (1 - ye^(xy))/(xe^(xy) - 1)

Answer

dy/dx = (1 + ye^(xy))/(xe^(xy) + 1)

Answer

dy/dx = (1 - ye^(xy))/(xe^(xy) + 1)

Answer

dy/dx = (1 + ye^(xy))/(xe^(xy) - 1)

Question 50

Find the derivative of y with respect to x for the equation ln(x² + y²) = 1.

Accepted Answer

dy/dx = -x/y

Answer

dy/dx = -y/x

Answer

dy/dx = x/y

Answer

dy/dx = y/x

Question 51

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

Accepted Answer

Isolate y in terms of x before differentiating.

Answer

Use the chain rule when differentiating terms involving y.

Answer

Differentiate both sides of the equation with respect to x.

Question 52

Find the points on the curve x² + xy + y² = 7 where the tangent line is horizontal.

Accepted Answer

(1, 2) and (2, 1)

Answer

(1, -2) and (-1, 2)

Answer

(1, -2) and (-2, 1)

Question 53

Find the derivative of y with respect to x for the equation x^y = y^x.

Accepted Answer

dy/dx = (y^x * ln(y) - xy^(x-1))/(x^y * ln(x) - xy^(y-1))

Answer

dy/dx = (y^x * ln(y) + xy^(x-1))/(x^y * ln(x) - xy^(y-1))

Answer

dy/dx = (y^x * ln(y) - xy^(x-1))/(x^y * ln(x) + xy^(y-1))