3.3 Rules for DifferentiationAKA “Shortcuts”

Review from 3.2•4 places derivatives do not exist:

▫Corner▫Cusp▫Vertical tangent (where derivative is

undefined)▫Discontinuity (jump, hole, vertical asymptote,

infinite oscillation)

• In other words, a function is differentiable everywhere in its domain if its graph is smooth and continuous.

3.2 Using the Calculator

•The calculator will find approximations for numerical derivatives.▫Ex.: Find the slope of the tangent line of f(x)

= x3 at x = 5.

3.2 Differentiability

•Theorem:▫If f has a derivative at x = a, then f is

continuous at x = a.▫If f is differentiable everywhere, it is also

continuous everywhere.

3.2 Intermediate Value Theorem for Derivatives

•If a and b are any two points in an interval on which f is differentiable, then f’ takes on every value between f’(a) and f’(b).

Derivatives of Constants

•Find the derivative of f(x) = 5.

h

xfhxfxf

h

)()(lim)('

0

hxf

h

55lim)('

0

0lim)('0

h

xf

0)(' xf

Derivative of a Constant:If f is the function with the constant value c, then,

0cdx

d

(the derivative of any constant is 0)

Power Rule•What is the derivative of f(x) = x3?

• From class the other day, we know f’(x) = 3x2.

• If n is any real number and x ≠ 0, then

1 nn nxxdx

d

In other words, to take the derivative of a term with a power, move the power down front and subtract 1 from the exponent.

Power Rule•Example:

▫What is the derivative of 5)( xxf

• Example:– What is the derivative of 4

1

)( xxf

45)(' xxf

43

4

1)('

xxf43

4

1

x

Power Rule•Example:

▫What is the derivative of x

xf1

)(

21

1)(x

xf

21

)( xxf Now, use power rule

23

2

1)(' xxf

23

2

1)('

xxf

Constant Multiple Rule

•Find the derivative of f(x) = 3x2.

h

xhxxf

h

22

0

3)(3lim)('

h

xhxhxxf

h

222

0

3)2(3lim)('

h

xhxhxxf

h

222

0

3363lim)('

0

xxf 6)('

Constant Multiple Rule:If u is a differentiable function of x and c is a constant, then

dx

duccu

dx

d)(

In other words, take the derivative of the function and multiply it by the constant.

Sum/Difference Rule•Find the derivative of f(x) = 3x2 + x

16)(' xxfSum/Difference Rule:If u and v are differentiable functions of x, then their sum and difference are differentiable at every point where u and v are differentiable. At such points,

dx

dv

dx

duvu

dx

d )(

In other words, if functions are separated by + or –, take the derivative of each term one at a time.

Example•Find where horizontal tangent occurs for

the function f(x) = 3x3 + 4x2 – 1.

xxxf 89)(' 2 A horizontal tangent occurs when the slope (derivative) equals 0.

089 2 xx0)89( xx

0x 089 x

9

8x

Example•At what points do the horizontal tangents

of f(x)=0.2x4 – 0.7x3 – 2x2 + 5x + 4 occur?

541.28.0)(' 23 xxxxfHorizontal tangents occur when f’(x) = 0

0541.28.0 23 xxxTo find when this polynomial = 0, graph it and find the roots.

862.1x 948.0x 539.3xSubstituting these x-values back into the original equation gives us the points (-1.862, -5.321), (0.948, 6.508), (3.539, -3.008)

Product Rule•If u and v are two differentiable functions,

then

dx

duv

dx

dvuuv

dx

d)(

Also written as:

'')( vuuvuvdx

d

In other words, the derivative of a product of two functions is “1st times the derivative of the 2nd plus the 2nd times the derivative of the 1st.”

Product Rule•Example:

Find the derivative of )18)(53( 4 xxxy

)3)(18()84)(53()(' 43 xxxxxf

324340202412)(' 434 xxxxxxf

43482015)(' 34 xxxxf

Quotient Rule•If u and v are two differentiable functions and

v ≠ 0, then

2vdxdvu

dxduv

v

u

dx

d

Also written as:

2

''

v

uvvu

v

u

dx

d

In other words, the derivative of a quotient of two functions is “low d-high minus high d-low all over low low.”

Quotient Rule•Example:

Find the derivative of 5

12)(

x

xxf

lowlow

dlowhighdhighlowxf

)('

2)5(

)1)(12()2)(5()('

x

xxxf

2)5(

12102)('

x

xxxf 5 if

)5(

112

xx

Higher-Order Derivatives• f’ is called the first derivative of f

dx

d

dx

dyor

• f'' is called the second derivative of f

2

2

or or '

dx

yd

dx

dy

dx

d

dx

dy

• f''' is called the third derivative of f

3

3

or ''

dx

yd

dx

dy

• f(n) is called the nth derivative of f

n

n

dx

yd

Higher-Order Derivatives•Example

Find the first four derivatives of 25 23 xxy

xxy 103' 2

106'' xy

6''' y

0)4( y