Rates of Change
(f(b)-f(a))/(b-a
Average Rate of Change
(Deltay)/(Deltax)
y=e^x+3sinx
Basic Rules
y=3/x^4
y'=-12x^-5
y'=-12/x^5
Differentiability implies continuity.
True
d/dy sinx
cos x
(f(x+h)-f(x))/(h)
Slope of the Secant Line
Average Rate of Change
y=root(3)4-x^2
Chain Rule
y=5xlnx
y'=5(lnx+1)
The instantaneous rate of change must equal the average rate of change somewhere in the interval.
Mean Value Theorem (MVT)
d/dy cscx
-csc x cot x
f'(x)=lim_(h->0) (f(x+h)-f(x))/(h)
Instantaneous Rate of Change
Slope of the Tangent Line
Derivative
y=sinx/e^(2x+3)
Quotient Rule
y=e^x/sinx
y'=((e^x)(sinx)-(e^x)(cosx))/(sinx)^2
y'=(e^x(sinx-cosx))/sin^2x
For a continuous function on a closed interval [a, b], you are guaranteed a height at least once.
Intermediate Value Theorem (IVT)
d/dy cos x
-sin x
f'(c)=lim_(x->c) (f(x)-f(c))/x-c
Alternate Definition
Derivative at a Point
y=xsin^(-1)x
Product Rule
y=cos^-1(2x+3)
y'=-2/sqrt(1-(2x+3)^2
y'=-2[1-(2x+3)^2]^(-1/2)
Continuity implies differentiability.
False
d/dy tan x
sec^2x
Daily Double
lim_(h->0) ((2+h)^2-2^2)/h
y=e^(xtanx)
Chain Rule
y=ln(5x^2+1)
y'=(10x)/(5x^2+1)
y'=10x(5x^2+1)^-1
When the derivative fails to exist.
Discontinuity, corner or cusp, vertical tangent
d/dy secx
secx tanx