mickey mouse math

haha

Limit definition of differentiability

1) f(x) must be continuous (if limit exists and value exists)

2) lim_x→c^-f’(x) = lim_x→c⁺f’(x)

3) f’(c)=L

4) lim_x→cf’(x)=f’(c)=L

every function that is differentiable is…

continuous

average rate of change is an approximation of…

instantaneous rate of change

limit definition of derivative

lim_h→0 ( f(x+h) - f(x) ) / h

lim_x→a ( f(x) - f(a) ) / x-a

d/dx (k) =

0

d/dx (xⁿ) =

nx^n-1

d/dx ( f(x) + or - g(x) ) =

f’(x) + or - g’(x)

d/dx (k f(x)) =

k f’(x)

d/dx (f(x)g(x)) =

f’(x)g(x) = g’(x)f(x)

d/dx (f(x)/g(x)) =

( f’(x)g(x) - g’(x)f(x) ) / (g(x))²

d/dx (sinx) =

cosx

d/dx (tanx) =

sec²x

d/dx (cosx) =

-sinx

d/dx (cscx) =

-cscxcotx

d/dx (cotx) =

-csc²x

d/dx (e^x) =

e^x

d/dx (a^x) =

(ln a) a^x

d/dx (ln x) =

1/x

d/dx (log_bx) =

(1/ln b)(1/x)

the derivative does not exist at a point where a function is…

not differentiable

to use the intermediate value theorem, the function must be…

continuous

equation of a tangent line should be in…

point slope form ONLY

Mean value theorem

If a function is differentiable at interval (a,b), then there is a c in (a,b) such that

f’(c)=(f(b) - f(a))/ b-a

or in other words, instantaneous rate of change equals average rate of change

derivative of the average rate of change is the…

instantaneous rate of change

intermediate value theorem→

functions

mean value theorem →

derivatives

d/dx (secx) =

secxtanx