Unit 2 Derivatives

general limit definition

lim_h→0 (f(x+h) - f(x)) /h

alternate limit definition

lim_x→c (f(x) - f(c)) / (x-c)

general limit definition

lim_h→0 (f(x+h) - f(x)) /h

alternate limit definition

lim_x→c (f(x) - f(c)) / (x-c)

differentiability reqs:

1. must be continuous

2. no sharp turns or vertical tangents

3. same derivative across a piecewise if needed

constant rule

if f(x) = c, f’(x) = 0

since f(x) is a straight line, the derivative (slope) is zero

power rule

if f(x) = xⁿ , then f’(x) = n(x)^n-1

derivative of exponential functions

e^x

derivative of natural log functions

1/x

d/dx sinx

cosx

d/dx tan x

sec²x

derivative of secx

secxtanx

d/dx cosx

-sinx

derivative of cotx

-csc²x

d/dx cscx

-cscxcotx

product rule f*g

h’(x) = f’(x)*g(x) + f(x)*g’(x)

quotient rule f/g

h’(x) = (f’(x)g(x) - f(x)g’(x)) / g(x)²