unit 2- differentiation definition and fundamental properties

d/dx cotx

-csc^2x OR -(csc)^2

d/dx cscx

-cscxcotx

dy/dx cosx

-sinx

d/dx (c)=

0

what three times is a function not differentiable?

1. sharp point 2. any point of discontinuity 3. vertical tangent line

d/dx (tan^-1 x)

1/(1+x^2)

d/dx loga(x)

1/xlna when using a common log 1. (ln(x))/(ln(a)) 2. cashew away in front 3. take the derivative

d/dx (cos^-1 x)

1/√(1-x^2)

d/dx (sin^-1 x)

1/√(1-x^2)

the derivative of anything that starts with a "C" is

NEGATIVE

d/dx (a^x)

a^x ln(a) a value raised to a function 1. rewrite 2. times the derivative of the power 3. times ln(n)

d/dx lnx

blah'/blah

(c f)'=

c f'(x) (stray cat)

d/dx sinx

cosx

d/dx eˣ

eˣ

Chain Rule f(g(x))

f'(g(x))g'(x) "derivative of the outer, leave the inner alone times derivative of the inner"

(f+g)'=

f'(x) +g'(x)

if y=f(x), then the derivative is defined as

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

Product rule of f(x)g(x)

f(x)g'(x)+g(x)f'(x) "derivative of the first times the second plus derivative of the second times the first"

what is the difference quotient?

f(x+h)-f(x)/h just another way of writing the slop formula because "h" is the difference between x1 and x2

Quotient rule of f(x)/g(x)

g(x)f'(x)-f(x)g'(x)/g(x)^2 "derivative of the top times the bottom minus derivative of the bottom times the top divided by the bottom squared"

d/dx (x^n)

nx^n-1 (power rule)

how do you find the value of f(x+h)?

plug (x+h) in for x in the formula and then solve with the given point

d/dx tanx

sec^2x OR (secx)^2

d/dx secx

secxtanx

what is the definition of a derivative at a specific point?

slope formula but as x gets closer to the first point

if f(t) is the amount of rain that had fallen t hours after midnight, explain the meaning of f^-1(10)=16.

that is inverse notation, so switch it. f^-1(10)= 16 means that f(16)= 10 so after 16 hours, 10 inches of rain had fallen

if y=f(x)...what is f'(a)?

the slope of the tangent line of y=f(x) at x=a m=f'(a)

what is the derivative of 2g(x)f(x)?

use the product rule but KEEP the constant WITH g(x)... dont pull it out. 2g'(x)f(x)+f'(x)2g(x)

how do you find the approximate derivative of a curve?

use the regular slope formula. it will give you the secant line between 2 points

what is the equation of a line tangent in terms of the derivative?

y-b= f'(a) (x-a)