Unit 2: Definition of Derivatives/Basic Derivatives

Average Rate of Change of f(x) on [a,b]

Average Rate of Change of f(x) on [a,b]

f(b)-f(a)/b-a

Instantaneous Rate of Change/Slope of tangent line/derivative (with h)

lim h->0 f(a+h)-f(a)/h

Instantaneous Rate of Change/Slope of tangent lint/derivative (with x)

lim x->a f(x)-f(a)/x-a

Definition of the Derivative

f'(x) = lim h->0 f(x+h)-f(x)/h

Equation of the Tangent Line

y-y1=m(x-x1); also point-slope form where m=derivative

Equation of the Normal Line

y-y1=1/m(x-x1)

Definition of a Vertical Tangent Line

lim h->0 f(c+h)-f(c)/h = +/- infinity

Derivative of a Constant

0

Derivative of f(x) = ax

a

Derivative of f(x) = x^n

nx^(n-1)

Derivative of Sinx

cosx

Derivative of cosx

-sinx

Derivative of e^x

e^x

Derivative of lnx

1/x

Derivative of a^x

a^xlna

Derivative of loga(x)

1/xlna

Product Rule

vdu + udv

Quotient Rule

vdu-udv/v^2

Derivative of tanx

sec^2x

Derivative of cotx

-csc^2x

Derivative of secx

secxtanx

Derivative of cscx

-cscxcotx