DerivativesDefinition of a DerivativePower RulePackage RuleProduct RuleQuotient RuleExponential Function and LogsTrigonometric Functions

Barbara WongPeriod 5

Definition of a Derivative

The slope of the tangent line to the graph of a function at a given point.

Mathematical Formula:

f ’(x) = Lim f (x+h) – f (x)

h0 h

Power Rule

y= x �

y’= � x-1�

Example: f(x) = 8x3 f '(x) = 24x2

Package Rule

d[a()� n] = na �n-1 d� dx dx

Example:

f(x) = 2(x2-1)2

f '(x) = 4(x2-1)1 (x2-1)’

= 8x(x2-1)

Product Rule

d ( � ) = � d+ d � dx dx dx

’� + �’

Example: f(x) = 3x ex f '(x) = 3ex + 3xex

= 3ex (x + 1)

Quotient Rule d(/� ) = d�dx � ddx

dx 2

’� - �’ 2

Example:

f (x) = x2 + 1

x3 f’(x) = (2x x3 ) – (x2 + 1) 3x2

x6

= 2 x4 – 3x4 – 3x2 = – x4 – 3x2

x6 x6

= – x2 – 3 x6

Rules for Simplifying Logs

1. ln( �) = ln() + ln(� )

2. ln � = ln() – ln(� )

3. ln � = ln()�Examples:

1. ln(2x) = ln(2) + ln(x)2. ln(x/2)= ln(x) – ln(2)3. lnx2 = 2lnx

Rules for Simplifying Natural Logs and Exponentials

1. ln(e) =

2. e ln =

(ex and lnx are inverse functions)

Examples:1. ln(e2) = 22. eln2 = 2

Derivatives of the Logarithm

& Exponential Functions

1. f(x) = ln � f’(x) = 1 (d)� �

Example: f(x) = ln x f '(x) = 1/x

2. f(x) = e � f’(x) = e �(d)� Example: f(x) = e3x

f '(x) = e3x 3 = 3e3x

Derivatives of Trigonometric Functions d(sin �) dx = cos � d � /dx

d(cos �) dx = -sin � d � /dx

d(tan �) dx = sec2 � d � /dx 

d(cot �) dx = -csc2 � d � /dx

d(sec �) dx = sec � tan � d �/dx 

d(csc �) dx = -csc � cot � d �/dx