DERIVATIVES

Review- 6 Differentiation Rules

For a function f(x) the instantaneous rate of change along the function is given by:

Which is called the derivative of f

f '(x) lim

x 0f (x x) f (x)

x

Definition of the Derivative

Notation:

3) find the Derivative of cos x

x

xxx

xxf

coscos

0

lim)(cos'

Trig Derivatives

x

dx

xdcos

sin

x

dx

xdsin

cos

x

dx

xd 2sectan

x

dx

xd 2csccot

xx

dx

xdtansec

sec

xx

dx

xdcotcsc

csc

The Power Rule

If n is a rational number, then the function

is differential and

d x n dx

nxn 1

f (x) x n

4) Find the derivative

2

25 63)(

x

xxxf

The Sum and Difference Rule

The derivative of a sum is the sum of the derivatives

Proof pg 111

If f (x) g(x) h(x) then

f '(x) g'(x) h'(x)

5) Find the derivative

xxxf sin5)( 3

The Product Rule

The product of two differentiable functions f and g is itself differentiable. Moreover, the derivative of fg is the first times the derivative of the second, plus the second times the derivative of the first

Proof pg 119

(fg) '(x) = f(x) g '(x) + g(x) f '(x)

6) Find the derivative

xxxf sin3)( 2

The Quotient Rule

The derivative of f g is given by:The denominator times the derivative of the numerator minus the numerator times the derivative of the denominator all divided by the denominator squared

Proof pg 121

f (x) n(x)

d(x)

d(x)n'(x) n(x)d'(x)

[d(x)]2

7) Find the derivative

53

12)(

2

x

xxf

The Chain Rule

If g(x) is differentiable then

Power rule times the derivative of inside

Proof pg 131

d[g(x)]k

dxk[g(x)]k 1

d(g(x))

dx

8) Find the derivative

3 22 1)( xxf

9) Find the derivative

xxf 4sin)( 3

HOME WORKPage 158 # 15-29 odd, 41, 45, 49, and 51, and Worksheet 6-B