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)