Chapter 3: Derivatives, Antiderivatives, and Indefinite Integrals

f(x)-f(c) / x - c

f’(c) = lim as x approaches c of f(x) - f(c) / x - c

the derivative of a function at a point equals the slope of the tangent line to the graph of the function at that point

both equal the instantaneous rate of change

x-increment, h

yes

yes

if f(x) = g(x) + h(x), where g and h are differentiable functions of x, then f’(x) = g’(x) + h’(x)

if f(x) = kg(x), where g is a differentiable function of x, then f’(x) = kg’(x), provided k is a constant

if f(x) = C, where C stands for a constant then f’(x) = 0 for all values of x

dy/dt

|v|

dv/dt

speeding up

slowing down

function g is an antiderivative (or indefinite integral) of function f if and only if g’(x) = f(x)

cosx

-sinx

if f(x) = g(h(x)), then f’(x) = g’(h(x) * h’(x)

to differentiate a composite function, differentiate the outside function with respect to the inside function (the inside function does not change, then multiply by the derivative of the inside function with respect to x

1

0

if

1. g(x) is less than or equal to h(x) for all x in a neighborhood of c where x does not equal c

2. limx—>c g(x) = lim x—>c h(x) = L

3. f is a function for which g(x) is less than or equal to f(x) is less than or equal to h(x) for all x in a neighborhood of c

then limx—>c f(x) = L

y = C + A cos B(x - D)

y = C

|A|

2pi * 1/B

D

e^x

logb(c^d) = d * logbc

logb(cd) = logbc + logbd

logbc/d = logbc - logbd

1/x

b^x ln b

lnx