Derivative and Integral Rules for AP Calculus flashcards

Question 1

For a constant c, what is d/dx (c)?

Accepted Answer

d/dx (c) = 0

Question 2

For differentiable functions f and g, what is d/dx (f(x) + g(x))?

Accepted Answer

d/dx (f(x) + g(x)) = d/dx (f(x)) + d/dx (g(x))

Question 3

For differentiable functions f and g, what is d/dx (f(x) − g(x))?

Accepted Answer

d/dx (f(x) − g(x)) = d/dx (f(x)) − d/dx (g(x))

Question 4

For a differentiable function f and a constant k, what is d/dx (k f(x))?

Accepted Answer

d/dx (k f(x)) = k d/dx (f(x))

Question 5

For a positive integer n, what is d/dx (xⁿ)?

Accepted Answer

d/dx (xⁿ) = n xⁿ⁻¹

Question 6

For a negative integer k, what is d/dx (xᵏ)?

Accepted Answer

d/dx (xᵏ) = k xᵏ⁻¹

Question 7

For differentiable functions f and g, what is d/dx (f(x) g(x))?

Accepted Answer

d/dx (f(x) g(x)) = d/dx (f(x))·g(x) + d/dx (g(x))·f(x)

Question 8

For differentiable functions f and g, what is d/dx (f(x)/g(x))?

Accepted Answer

d/dx (f(x)/g(x)) = (d/dx (f(x))·g(x) − d/dx (g(x))·f(x)) / (g(x))²

Question 9

For a positive integer n and an arbitrary integer m, what is d/dx (x^(m/n))?

Accepted Answer

d/dx (x^(m/n)) = (m/n) x^((m/n)−1)

Question 10

Let f and g be functions, and let h(x) = f(g(x)). At each x where g is differentiable at x and f is differentiable at g(x), what is h′(x) (chain rule)?

Accepted Answer

h′(x) = f′(g(x)) g′(x)

Question 11

If y is a function of u and u is a function of x, what is dy/dx (chain rule, Leibniz form)?

Accepted Answer

dy/dx = (dy/du)·(du/dx)

Question 12

For all values of x for which the derivative is defined, if h(x) = (g(x))ⁿ, what is h′(x)?

Accepted Answer

h′(x) = n (g(x))ⁿ⁻¹ g′(x)

Question 13

What is d/dx (sin x)?

Accepted Answer

d/dx (sin x) = cos x

Question 14

What is d/dx (cos x)?

Accepted Answer

d/dx (cos x) = −sin x

Question 15

What is d/dx (tan x)?

Accepted Answer

d/dx (tan x) = sec²x

Question 16

What is d/dx (cot x)?

Accepted Answer

d/dx (cot x) = −csc²x

Question 17

What is d/dx (sec x)?

Accepted Answer

d/dx (sec x) = sec x tan x

Question 18

What is d/dx (csc x)?

Accepted Answer

d/dx (csc x) = −csc x cot x

Question 19

What is d/dx (eˣ)?

Accepted Answer

d/dx (eˣ) = eˣ

Question 20

For x > 0, what is d/dx (ln x)?

Accepted Answer

d/dx (ln x) = 1/x

Question 21

Let b > 0 and b ≠ 1. What is d/dx (log_b x)?

Accepted Answer

d/dx (log_b x) = 1/(x ln b)

Question 22

Let b > 0 and b ≠ 1. What is d/dx (bˣ)?

Accepted Answer

d/dx (bˣ) = bˣ ln b

Question 23

What is d/dx (sin⁻¹ x), also written arcsin x?

Accepted Answer

d/dx (sin⁻¹ x) = 1/√(1 − x²)

Question 24

What is d/dx (cos⁻¹ x), also written arccos x?

Accepted Answer

d/dx (cos⁻¹ x) = −1/√(1 − x²)

Question 25

What is d/dx (tan⁻¹ x), also written arctan x?

Accepted Answer

d/dx (tan⁻¹ x) = 1/(1 + x²)

Question 26

When is a function F an antiderivative of a function f?

Accepted Answer

When F′(x) = f(x) for all x in the domain of f

Question 27

If F is an antiderivative of f over an interval I, what is the most general form of an antiderivative of f over I?

Accepted Answer

F(x) + C, for a constant C

Question 28

If F is an antiderivative of f, what is the indefinite integral ∫ f(x) dx?

Accepted Answer

∫ f(x) dx = F(x) + C, the most general antiderivative of f

Question 29

For n ≠ −1, what is ∫ xⁿ dx?

Accepted Answer

∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C

Question 30

For a constant k, what is ∫ k dx?

Accepted Answer

∫ k dx = k x + C

Question 31

What is ∫ 1/x dx?

Accepted Answer

∫ 1/x dx = ln|x| + C

Question 32

What is ∫ eˣ dx?

Accepted Answer

∫ eˣ dx = eˣ + C

Question 33

What is ∫ aˣ dx?

Accepted Answer

∫ aˣ dx = aˣ/(ln a) + C

Question 34

What is ∫ cos x dx?

Accepted Answer

∫ cos x dx = sin x + C

Question 35

What is ∫ sin x dx?

Accepted Answer

∫ sin x dx = −cos x + C

Question 36

What is ∫ sec²x dx?

Accepted Answer

∫ sec²x dx = tan x + C

Question 37

What is ∫ csc²x dx?

Accepted Answer

∫ csc²x dx = −cot x + C

Question 38

What is ∫ sec x tan x dx?

Accepted Answer

∫ sec x tan x dx = sec x + C

Question 39

What is ∫ csc x cot x dx?

Accepted Answer

∫ csc x cot x dx = −csc x + C

Question 40

What is ∫ 1/(1 + x²) dx?

Accepted Answer

∫ 1/(1 + x²) dx = tan⁻¹ x + C

Question 41

What is ∫ 1/√(1 − x²) dx?

Accepted Answer

∫ 1/√(1 − x²) dx = sin⁻¹ x + C

Question 42

Let F and G be antiderivatives of f and g. What is ∫ (f(x) ± g(x)) dx?

Accepted Answer

∫ (f(x) ± g(x)) dx = F(x) ± G(x) + C

Question 43

Let F be an antiderivative of f, and let k be any real number. What is ∫ k f(x) dx?

Accepted Answer

∫ k f(x) dx = k F(x) + C

Question 44

If f(x) is continuous over [a, b], what is the average value of f on [a, b]?

Accepted Answer

f_ave = (1/(b − a)) ∫_a^b f(x) dx

Question 45

If f(x) is continuous over [a, b] and F(x) = ∫_a^x f(t) dt, what is F′(x), and over what interval (Fundamental Theorem of Calculus, Part 1)?

Accepted Answer

F′(x) = f(x) over (a, b)

Question 46

If f is continuous over [a, b] and F(x) is any antiderivative of f(x), what is ∫_a^b f(x) dx (Fundamental Theorem of Calculus, Part 2)?

Accepted Answer

∫_a^b f(x) dx = F(b) − F(a)

Question 47

Let u = g(x), let g′ be continuous over an interval [a, b], and let f be continuous over the range of u = g(x). What is ∫_a^b f(g(x)) g′(x) dx (substitution)?

Accepted Answer

∫_a^b f(g(x)) g′(x) dx = ∫_{g(a)}^{g(b)} f(u) du

Question 48

Let u = f(x) and v = g(x) be functions with continuous derivatives. What is ∫ u dv (integration by parts)?

Accepted Answer

∫ u dv = u v − ∫ v du

Question 49

Let u = f(x) and v = g(x) be functions with continuous derivatives on [a, b]. What is ∫_a^b u dv (integration by parts)?

Accepted Answer

∫_a^b u dv = u v |_a^b − ∫_a^b v du

Derivative and Integral Rules for AP Calculus flashcards

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Derivative and Integral Rules for AP Calculus flashcards

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