derivatives

t/f: if a function is differentiable then it is continuous

true

t/f: if continuous then differentiable

false

t/f: if a function is NOT continuous then NOT differentiable

True

t/f: if NOT differentiable then NOT continuous

false

what graphical features make a function not differentiable

a cusp/sharp corner, a discontinuity, vertical tangent (steep downward hill)

d/dx(sinx)

cosx

d/dx(cosx)

-sinx

d/dx(tanx)

sec²x

d/dx(cotx)

-csc²x

d/dx(secx)

(secx)(tanx)

d/dx(cscx)

-csc(x)cot(x)

d/dx sqrt(x)

1/2sqrt(x)

d/dx e^x

e^x

d/dx lnx

1/x

d/dx b^x

(b^x)(lnb)

d/dx logb(x)

1/(x)(lnb)

AROC equation

y2-y1/x2-x1

definition of a derivative at a point

f’(c)=limx→c f(x)-f(c)/x-c

definition of a derivative as a function

f’(x)= lim h→0 f(x+h)-f(x)/h

power rule definition if f(x)=ax^n

f’(x)=(n•a)x^n-1

power rule example: 3x³

9x²

product rule definition if h(x)=f(x)•g(x)

h’(x)=f’(x)•g(x)+f(x)•g’(x)

product rule example (3x²)(sinx)

6x•sinx+3x²•cosx

quotient rule if h(x)=f(x)/g(x)

h’(x)=f’(x)•g(x)-f(x)•g’(x)/g(x)²

chain rule definition if h(x)=f(g(x))

h’(x)=f’(g(x))•g’(x)

chain rule example: sin²x

2(sinx)•cosx

quotient rule example x²/5x

2x•5x-x²•5/5x²