5.3 differentiation rules + 5.4 graphing derivatives

constant multiple

(cf’)(x)=c'(f’(x))

• f(x)=10² → f’(10x²) → 10f’(x²) → limit defintion

• limit exists

first derivative test

1. find derivative

2. find critical points/undefined points

3. create a sign chart with intervals

4. find extrema (where f is defined + f’ changes signs) → local/relative

• positive to negative is max

• negative to positive is min

how to find absolute max and min

is either the endpoints of domain or on domain

1. find f( c) for every critical value of f’(c )

2. evaluate f(x) at endpoints → f(a) and f(b)

3. largest f(x) out of all potential ones is abs max and the smallest is abs min

how to do the second derivative test

1. find cv of f’(x) when it equals 0 or is undefined

2. find second derivative

3. sub cv into f’’(x)

4. if positive, up. if negative, down. if 0, inconclusive. use first derivative tes

f’(x)=0, f”(x)<0

f(x) has local max at x

f’(x)=0, f”(x)>0

f(x) has local min at x

f’(x)=0, f”(x)=0

f” not working

point of inflection

a change in f(x)’s concavity (f” would equal 0 at this point)

horizontal point of inflection

stationary point of inflection

how to sketch f’ or f’’ from f

1. label with axes

2. x-intercepts

3. turning points

4. asymptotes

5. point of inflection

turning points

when f’ changes signs. when poi too