Calculus Study Guide: Limits, Derivatives, and Growth

Limits

Evaluate behavior of functions as inputs approach values.

L'Hôpital's Rule

Apply for indeterminate forms in limits.

Taylor Series

Use for small values of x.

Trigonometric Limits

Specific limits involving trigonometric functions.

Power Rule

d/dx x^n = nx^n-1

Chain Rule

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

Product Rule

d/dx[f(x)g(x)]=f'(x)g(x)+f(x)g'(x)

Quotient Rule

d/dx (f(x) / g(x)) = (f'(x)g(x) - f(x)g'(x)) / (g(x))^2

Common Derivatives

Standard derivatives for basic functions.

Logarithmic Equations

Equations involving logarithmic expressions.

Exponential Equations

Equations involving exponential expressions.

Relative Growth Rate

Rate of change of population over time.

E.coli Growth Rate

Determine growth rate for bacteria doubling time.

Natural Logarithm

Logarithm base e, used for solving equations.

Inverse Functions

Functions that reverse the effect of another function.

Higher-Order Derivatives

Derivatives beyond the first derivative.

Second Derivative

Derivative of the first derivative, indicates concavity.

Nth Derivative

f^n(x) = d^n/dx^n f(x)

Basic Trig Derivatives

Derivatives of sine, cosine, tangent functions.

Logarithmic Derivatives

Derivatives of logarithmic functions.

Exponential Derivatives

Derivatives of exponential functions.

d/dx e^x =

e^x

d/dx sin^-1 x=

1/\sqrt(1-x^2)

d/dx cos^-1 x=

- 1/\sqrt(1-x^2)

d/dx tanh x=

sech^2 x

ln e^x=

x

e^ln x=

x

a^lnb=

b^ln a

solving e^x=k

x=ln k

solving e^e^x = 2

take the natural logarithm twice

Exponential Growth Formula

Y(t)=Y0e^rt

Y0 =

initial population

r=

relative growth rate

t=

time

determine r using..

known population changes

solve for Y(t)..

at given t

to find the growth of rate..

differentiate

solve for t when

Y(t)=k for a specific value

use logarithms to..

simplify exponentials before differentiating

apply the chain rule when..

differentiating composite functions

Constant Rule for Derivatives

d/dx C=0

Constant Multiple Rule

d/dx[Cf(x)]=C'f'(x)

Sum & Difference Rules

d/dx [f(x)+/- g(x)]=f/x+/-g'(x)

Product Rule

d/dx[uv]=u'v+uv'

Quotient Rule

d/dx(u/v)=(u'v-uv')/v^2

Chain Rule

d/dxf(g(x))=f'g(x))g'(x)

d/dx sin x=

cos x

d/dx cos x=

-sin x

d/dx tan x=

sec^2 x

d/dx cot x=

-csc^2 x

d/dx sec x=

sec x tan x

d/dx csc x=

-csc x cot x

d/dx ln x=

1/x

d/dx loga x=

1/x ln a

d/dx tan^-1 x=

1/1+x^2

d/dx cot^-1 x=

-1/1+x^2

d/dx sec^-1 x=

1/|x| \sqrt(x^2-1)

d/dx csc^-1 x= -1/|x| \sqrt(x^2-1)

f''(x)=

d^2/dx^2 f(x)

f^n (x)=

d^n/dx^n f(x)