Section 3.3a

*Power, Sum, and Difference Rules, Higher-Order Derivatives

The “Do Now”Find the derivative of 5f x

0

limh

f x h f xf x

h

0

5 5limh h

0lim0h

0

Does this make sense graphically???

The “Do Now”Find the derivative of 3 22f x x x

3 2 2 3 2 2 3 2

0

2 6 6 2 2 2limh

x x h xh h x xh h x xh

3 2 3 2

0

2 2limh

x h x h x xf x

h

Let’s see some patterns so we can generalizesome easier rules for finding these derivatives…

2 2

0lim 6 6 2 2h

x xh h x h

26 2x x

Suppose is a function with a constant value c. f x c

Then 0

limh

f x h f xf x

h

0limh

c ch

0lim0h

0

Rule 1: Derivative of a Constant Function

fIf is the function with the constant value c, then

0df d cdx dx

If , then thedifference quotient for is:

nf x xf n nx h x

h

1 2 2 1n n n n n na b a b a a b ab b

1 2 2 1n n n nh x h x h x x h x x

h

Recall the algebraic identity:

a x h Here, we will let and :b x

1 2 2 1n n n nx h x h x x h x x

1

0limn n

h

f x h f xd x nxdx h

What happens when we let h 0?

Rule 2: Power Rule for Integer Powers of xnIf is an integer, then

1n nd x nxdx

The Power Rule says: To differentiate x , multiplyby n and subtract 1 from the exponent.

n

Suppose we have a function comprised of a differentiable functionof x multiplied by a constant: f x c u x

Then 0

limh

cu x h cu xdf x cudx h

0

limh

u x h u xc

h

ducdx

Rule 3: The Constant Multiple RuleuIf is a differentiable function of x and c is a constant, then

d ducu cdx dx

The constant “goesalong for the ride”

Returning to the “Do Now”Find the derivative of 5f x

Find the derivative of 3 22f x x x

5 0df df xdx dx

3 2 22 6 2df df x x x x xdx dx

What if we have a function comprised of a sum of otherdifferentiable functions? f x u x v x The derivative:

0

limh

u x h v x h u x v xf x

h

0

limh

u x h u x v x h v xh h

0 0lim limh h

u x h u x v x h v xh h

du dvdx dx

Rule 4: The Sum and Difference RuleIf u and v are differentiable functions of x, then their sum anddifference are differentiable at every point where u and v aredifferentiable. At such points,

d du dvu vdx dx dx

The derivative of a sum (or difference) isthe sum (or difference) of the derivatives

Guided Practice

Finddpdt

if3 2 56 16

3p t t t

3 2 56 163

dp d d d dt t tdt dt dt dt dt

Take each term separately:

2 53 6 2 03

t t

2 53 123

t t

Guided Practice

Find y if1

4 2 734 5 2 145xy x x x

5 2 1 634 4 1 5 2 2 7 05

y x x x x

5 2 6316 10 145

x x x x

With no negative exponents:

62 5

3 1614 105

x xx x

Second and Higher Order DerivativesThe derivative is called the first derivative of ywith respect to x. The first derivative may itself be a differentiablefunction of x. If so, its derivative

y dy dx

2

2

dy d dy d yydx dx dx dx

is called the second derivative of y with respect to x. If(“y double-prime”) is differentiable, its derivative

y

3

3

dy d yydx dx

is called the third derivative of y with respect to x. The patterncontinues…

Second and Higher Order DerivativesHowever, the multiple-prime notation gets too cumbersome afterthree primes. We use

1n ndy ydx

to denote the n-th derivative of y with respect to x. (We alsouse .)n nd y dx

nyNote: Do not confuse with the n-th power of y, which isny

Guided PracticeFind the first four derivatives of .

3 25 2y x x

First derivative: 23 10y x x

Second derivative: 6 10y x

Third derivative: 6y

Fourth derivative: 4 0y

Guided PracticeDoes the curve have any horizontaltangents? If so, where?

4 22 2y x x

If there are any horizontal tangents, they occur where the slopedy/dx is zero… 4 22 2dy d x x

dx dx 34 4x x

34 4 0x x Solve the equation dy/dx = 0 for x:

24 1 0x x

0, 1,1x 4 1 1 0x x x How can we support

our answer graphically?