AP Calculus: Derivatives flashcards

Question 1

State the limit definition of the derivative of f at a, using h.

Accepted Answer

f′(a) = lim as h → 0 of [f(a + h) − f(a)]/h

Question 2

What is the difference quotient of f at a with increment h?

Accepted Answer

[f(a + h) − f(a)]/h

Question 3

What is acceleration in terms of velocity?

Accepted Answer

The derivative (rate of change) of velocity

Question 4

What is the speed of an object with velocity v(t)?

Accepted Answer

|v(t)|, the absolute value of velocity

Question 5

When is a function differentiable at a?

Accepted Answer

When f′(a) exists

Question 6

What is differentiation?

Accepted Answer

The process of taking a derivative

Question 7

What does differentiability at a point imply about continuity there?

Accepted Answer

It is continuous at that point

Question 8

Name three ways a function can fail to be differentiable at a point.

Accepted Answer

It is not continuous there, it has a vertical tangent there, or its graph has a sharp corner or cusp

Question 9

Where the tangent line to f is horizontal, what is f′(x)?

Accepted Answer

f′(x) = 0

Question 10

What is the derivative of a constant c?

Accepted Answer

0

Question 11

State the power rule for an integer n.

Accepted Answer

d/dx x^n = n x^(n − 1)

Question 12

State the product rule.

Accepted Answer

d/dx [f(x)g(x)] = f′(x)g(x) + g′(x)f(x)

Question 13

State the quotient rule.

Accepted Answer

d/dx [f(x)/g(x)] = [f′(x)g(x) − g′(x)f(x)]/(g(x))^2

Question 14

State the chain rule for h(x) = f(g(x)).

Accepted Answer

h′(x) = f′(g(x)) g′(x)

Question 15

How is the chain rule written in Leibniz notation?

Accepted Answer

dy/dx = (dy/du)(du/dx)

Question 16

What is d/dx (sin x)?

Accepted Answer

cos x

Question 17

What is d/dx (cos x)?

Accepted Answer

−sin x

Question 18

What is d/dx (tan x)?

Accepted Answer

sec^2 x

Question 19

What is d/dx (sec x)?

Accepted Answer

sec x tan x

Question 20

What is d/dx (cot x)?

Accepted Answer

−csc^2 x

Question 21

What is d/dx (csc x)?

Accepted Answer

−csc x cot x

Question 22

By the inverse function theorem, what is (f^−1)′(x)?

Accepted Answer

1/f′(f^−1(x)), provided f is differentiable and f′(f^−1(x)) ≠ 0

Question 23

What is d/dx (sin^−1 x)?

Accepted Answer

1/√(1 − x^2)

Question 24

What is d/dx (tan^−1 x)?

Accepted Answer

1/(1 + x^2)

Question 25

What is d/dx (e^g(x))?

Accepted Answer

e^g(x) · g′(x)

Question 26

What is d/dx (ln g(x))?

Accepted Answer

g′(x)/g(x)

Question 27

What is a higher-order derivative?

Accepted Answer

The derivative of a derivative (second, third, up to nth derivative)

Question 28

What is marginal cost?

Accepted Answer

The derivative of the cost function

Question 29

What is the population growth rate?

Accepted Answer

The time derivative of the population

AP Calculus: Derivatives flashcards

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