Memorize the derivative and antiderivative formulas given in Table 1. In this table, c and n denote any constants, while b denotes any positive constant except for b = 1. The function G denotes an antiderivative of the function g. d Definition. G is called an antiderivative of g on an interval I if G(x)=g(x) for all x ∈ I. dx (See Stewart p. 350.) Table 1. Basic Derivative & Antiderivative Formulas

f(x) f (x) g(x) G(x)

c 0 0 c

x 1 1 x

xn+1 xn nxn−1 xn (n = −1) n +1

ex ex ex ex

bx bx bx ln b bx ln b

1 1 ln |x| ln |x| x x

1 logb |x| ∗ ∗ x ln b

sin x cos x cos x sin x

cos x − sin x sin x − cos x

tan x sec2 x sec2 x tan x

cot x − csc2 x csc2 x − cot x

sec x sec x tan x sec x tan x sec x

csc x − csc x cot x csc x cot x − csc x

1 2

f(x) f (x) g(x) G(x)

1 1 sin−1 x √ √ sin−1 x 1 − x2 1 − x2

1 cos−1 x −√ ∗ ∗ 1 − x2

1 1 tan−1 x tan−1 x 1+x2 1+x2

1 cot−1 x − ∗ ∗ 1+x2

1 1 sec−1 x √ √ sec−1 x x x2 − 1 x x2 − 1

1 csc−1 x − √ ∗ ∗ x x2 − 1

Let f and g denote functions that are differentiable on an interval I.Letc denote a constant. Then for x ∈ I we have the following differentiation rules: Constant Multiple Rule d [cf(x)] = cf (x) dx In words: The derivative of a constant times a function is equal to the constant times the derivative of the function. Sum & Difference Rules d [f(x) ± g(x)] = f (x) ± g(x) dx In words: The derivative of a sum (difference) is equal to the sum (difference) of the derivatives. Product Rule d [f(x)g(x)] = f(x)g(x)+g(x)f (x) dx In words: The derivative of a product of two functions is equal to the first function times the derivative of the second function plus the second function times the derivative of the first function. Quotient Rule   d f(x) g(x)f (x) − f(x)g(x) = (provided g(x) =0) dx g(x) [g(x)]2 Mnemonic: Low d’high minus high d’low over the square of what’s below. 3

Chain Rule d f(g(x)) = f (g(x))g(x) dx In words: The derivative of a composite function is equal to the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Chain Rule in Leibniz notation If y = f(g(x)) is rewritten in the form y = f(u), where u = g(x), then dy dy du = dx du dx Power Chain Rule (i.e., the Power Rule combined with the Chain Rule) Let n be a real number. Let u = g(x) be a differentiable function. d d d d du du [g(x)]n = n[g(x)]n−1 g(x)or un = un · = nun−1 dx dx dx du dx dx

The derivatives of the other functions listed in Table 1 generalize in a similar way, such as d du d 1 du d du d 1 du eu = eu , ln |u| = , sin u =cosu , tan−1 u = dx dx dx u dx dx dx dx 1+u2 dx