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Practice Set B — Differentiation: Composite, Implicit, and Inverse Functions

20 questions • 30 minutes • auto-graded with full solutions

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A second auto-graded set covering Differentiation: Composite, Implicit, and Inverse Functions.

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Multiple Choice

Question 1

Let f(x) = x^3 + x + 1. If g is the inverse of f, what is g'(1)?

Solution: f(0) = 1, so g(1) = 0. f'(x) = 3x^2 + 1, so f'(0) = 1. Then g'(1) = 1/f'(0) = 1/1 = 1.

Question 2

Suppose f(1) = 4 and f'(1) = 2. If g = f^(-1), what is g'(4)?

Solution: g'(4) = 1 / f'(g(4)) = 1 / f'(1) = 1/2, since g(4) = 1.

Question 3

A differentiable function f has an inverse g. A table shows f(3) = 7 and f'(3) = 1/2. What is g'(7)?

Solution: g'(7) = 1 / f'(g(7)) = 1 / f'(3) = 1 / (1/2) = 2.

Question 4

If f(x) = e^x, and g is its inverse, what is g'(x)?

Solution: The inverse of e^x is ln(x), whose derivative is 1/x. Equivalently, g'(x) = 1/f'(g(x)) = 1/e^(ln x) = 1/x.

Question 5

What is d/dx [ arctan(2x) ]?

Solution: By the Chain Rule, d/dx arctan(u) = u'/(1 + u^2). With u = 2x, the result is 2 / (1 + 4x^2).

Question 6

What is d/dx [ arcsin(x^2) ]?

Solution: By the Chain Rule, d/dx arcsin(u) = u'/sqrt(1 - u^2). With u = x^2, u' = 2x and u^2 = x^4, giving 2x / sqrt(1 - x^4).

Question 7

What is the value of d/dx [ arctan(x) ] at x = 1?

Solution: d/dx arctan(x) = 1/(1 + x^2). At x = 1, this equals 1/(1 + 1) = 1/2.

Question 8

What is d/dx [ arcsin(x/3) ]?

Solution: d/dx arcsin(u) = u'/sqrt(1 - u^2) with u = x/3, u' = 1/3. This gives (1/3)/sqrt(1 - x^2/9) = (1/3)/(sqrt(9 - x^2)/3) = 1/sqrt(9 - x^2).

Question 9

What is d/dx [ arccos(2x) ]?

Solution: d/dx arccos(u) = -u'/sqrt(1 - u^2). With u = 2x, u' = 2, giving -2 / sqrt(1 - 4x^2).

Question 10

What is d/dx [ 3 arcsin(x) ]?

Solution: The constant multiple stays: 3 * d/dx arcsin(x) = 3 / sqrt(1 - x^2).

Question 11

What is d/dx [ x^2 sin(x) ]?

Solution: By the Product Rule: (2x) sin(x) + x^2 (cos x) = 2x sin(x) + x^2 cos(x).

Question 12

What is d/dx [ sin(x) / x ]?

Solution: By the Quotient Rule: (x * cos(x) - sin(x) * 1) / x^2 = (x cos(x) - sin(x)) / x^2.

Question 13

What is d/dx [ x e^x ]?

Solution: By the Product Rule: (1) e^x + x e^x = e^x(1 + x) = e^x(x + 1).

Question 14

Which rule is most appropriate to differentiate f(x) = (3x + 2)^10?

Solution: The function is a composite (an inner linear function raised to a power), so the Chain Rule is used: 10(3x + 2)^9 * 3.

Question 15

What is d/dx [ (x^2 + 1)(x^3 - 2) ]?

Solution: Product Rule: (2x)(x^3 - 2) + (x^2 + 1)(3x^2) = 2x^4 - 4x + 3x^4 + 3x^2 = 5x^4 + 3x^2 - 4x.

Question 16

What is d/dx [ tan(x) / x ]?

Solution: Quotient Rule: (x * sec^2(x) - tan(x) * 1) / x^2 = (x sec^2(x) - tan(x)) / x^2.

Question 17

What is d/dx [ sqrt(x) sin(x) ]?

Solution: Product Rule with d/dx sqrt(x) = 1/(2 sqrt(x)): (1/(2 sqrt(x))) sin(x) + sqrt(x) cos(x).

Question 18

To differentiate f(x) = cos(3x) / e^x, which combination of rules is required?

Solution: The function is a quotient (needs the Quotient Rule), and the numerator cos(3x) is composite (needs the Chain Rule).

Question 19

If f(x) = e^(2x), what is f''(x)?

Solution: f'(x) = 2e^(2x), and f''(x) = 4e^(2x).

Question 20

If f(x) = cos(x), what is the fourth derivative f''''(x)?

Solution: Derivatives cycle: f' = -sin(x), f'' = -cos(x), f''' = sin(x), f'''' = cos(x).