2.1 Tangents and Derivatives at a Point

Finding a Tangent to the Graph of a Function

To find a tangent to an arbitrary curve y=f(x) at a point P(x0,f(x0)), we

• Calculate the slope of the secant through P and a nearby point

Q(x0+h, f(x0+h)).

• Then investigate the limit of the slope as h0.

Slope of the Curve

If the previous limit exists, we have the following definitions.

Reminder: the equation of the tangent line to the curve at P is Y=f(x0)+m(x-x0) (point-slope equation)

Example

(a) Find the slope of the curve y=x2 at the point (2, 4)?

(b) Then find an equation for the line tangent to the curve there.

Solution (a)

(b) The equation is

y=4+4(x-2), that is, y=4x-4.

0 0

0 0

2 2 2

0 0

2

0 0

( ) ( ) (2 ) (2)lim lim

(2 ) 2 4 4 4lim lim

4lim lim(4 )

4

h h

h h

h h

f x h f x f h fm

h h

h h h

h h

h hh

h

Derivative of a Function f at a Point x0

The expression

is called the difference quotient of f at x0 with increment h.

If the difference quotient has a limit as h approaches zero, that limit is named below.

( ) ( )f x h f x

h

2.2 The Derivative as a Function

We now investigate the derivative as a function derived from f byConsidering the limit at each point x in the domain of f.

If f’ exists at a particular x, we say that f is differentiable (has a derivative) at x. If f’ exists at every point in the domain of f, we call fis differentiable.

Alternative Formula for the Derivative

An equivalent definition of the derivative is as follows. (let z = x+h)

Calculating Derivatives from the Definition

The process of calculating a derivative is called differentiation. It can be denoted by

Example. Differentiate

Example. Differentiate for x>0.

'( ) ( )d

f x or f xdx

( )f x x

2( )f x x

Notations

'( ) ' ( ) ( )( ) [ ( )]x

dy df df x y f x D f x D f x

dx dx dx

'( ) | | ( ) |x a x a x a

dy df df a f x

dx dx dx

There are many ways to denote the derivative of a function y = f(x). Some common alternative notations for the derivative are

To indicate the value of a derivative at a specified number x=a, we use the notation

If a function f is differentiable on an open interval (finite or infinite) if it has a derivative at each point of the interval.

It is differentiable on a closed interval [a, b] if it is differentiable on the interior (a, b) and if the limits

exist at the endpoints.

0

( ) ( )limh

f a h f a

h

0

( ) ( )limh

f b h f b

h

Differentiable on an Interval; One-Sided Derivatives

Right-hand derivative at a

Left-hand derivative at b

A function has a derivative at a point if and only if the left-hand and right-hand derivatives there, and these one-sided derivatives are equal.

When Does A Function Not Have a Derivative at a Point

A function can fail to have a derivative at a point for several reasons, such as at points where the graph has

1. a corner, where the one-sided derivatives differ.

2. a cusp, where the slope of PQ approaches from one side and - from the other.

3. a vertical tangent, where the slope of PQ approaches from both sides or approaches - from both sides.

4. a discontinuity.

Differentiable Functions Are Continuous

Note: The converse of Theorem 1 is false. A function need not have a derivative at a point where it is continuous.

For example, y=|x| is continuous at everywhere but is not differentiable at x=0.

2.3 Differentiation Rules

The Power Rule is actually valid for all real numbers n.

5 4[ ] 5 ,dx x

dx

1 1 1 22

1 1[ ] [ ] 1

d dx x x

dx x dx x

1 1

2 21 1

[ ] [ ]2 2

d dx x x

dx dx x

3 2[ ] ( 3)e edx e x

dx

Example.

7 7 6 6[3 ] 3 [ ] 3(7 ) 21d d

x x x xdx dx

11 11 10 10

2 3 32

[ ] [ ] (11 ) 11

[ ] [ ] ( 2 ) 2

d dx x x x

dx dxd d

x x xdx x dx

Example.

11 11 10 10[ ] [ ] (11 ) 11d d

x x x xdx dx

2 3 32

[ ] [ ] ( 2 ) 2d d

x x xdx x dx

Note: ( ) ( 1 ) 1 ( )d d d du

u u udx dx dx dx

8 9 8 9 7 10 7 10[3 2 ] [3 ] [2 ] 24 2( 9) 24 18d d d

x x x x x x x xdx dx dx

1 3[10 6 ] [10] [6 ] 0 6( )

2

d d dx x

dx dx dx x x

Example.

Example: 3 2Find if (2 2)(6 3 ).dy

y x x xdx

Solution: 3 2

3 2 2 3

3 2 2

4 3 4 3

4 3

[(2 2)(6 3 )]

(2 2) [6 3 ] (6 3 ) [2 2]

(2 2)(12 3) (6 3 )(6 )

24 6 24 6 36 18

60 24 24 6

dy dx x x

dx dxd d

x x x x x xdx dx

x x x x x

x x x x x

x x x

Example

Example: 3 22 4

Find '( ) if .5

x xy x y

x

Solution: 3 2

3 2 3 2

2

2 3 2

2

3 2 2 3 2

2

3 2

2

2 4[ ]

5

( 5) [2 4] (2 4) [ 5]

( 5)

( 5)(6 2 ) (2 4)(1)

( 5)

(6 2 30 10 ) (2 4)

( 5)

4 31 10 4

( 5)

dy d x x

dx dx xd d

x x x x x xdx dx

x

x x x x x

x

x x x x x x

x

x x x

x

Example

The derivative f’ of a function f is itself a function and hence may have a derivative of its own.

If f’ is differentiable, then its derivative is denoted by f’’. So f’’=(f’)’ and is called the second derivative of f.

Similarly, we have third, fourth, fifth, and even higher derivatives of f.

Higher derivatives

22 2

2

'''( ) ( ) '' ( )( ) [ ( )]x

d y d dy dyf x y D f x D f x

dx dx dx dx

A general nth order derivative can be denoted by

( ) ( 1)n

n n nn

d d yy y D y

dx dx

Example: 3 2 y 4 2 6, thenIf x x x

2

(4)

( )

' 12 2 2

'' 24 2

''' 24

0

0( 4)n

y x x

y x

y

y

y n

2.5 Derivatives of Trigonometric Functions

Example: Find if cos .dy

y x xdx

Solution: [ cos ]

[cos ] cos [ ]

( sin ) cos (1)

cos sin

dy dx x

dx dxd d

x x x xdx dxx x x

x x x

Example

Example: cos

Find if .1 sin

dy xy

dx x

2

2

2 2

2

2

(1 sin ) [cos ] cos [1 sin ]=

(1 sin )

(1 sin )( sin ) cos (cos )

(1 sin )

sin sin cos

(1 sin )

sin 1

(1 sin )

d dx x x xdy dx dx

dx x

x x x x

x

x x x

x

x

x

Solution:

Example

Example: Find '' if ( ) tan .y f x x

2

2

2

' [tan ] sec

'' [sec ] [sec sec ]

sec [sec ] sec [sec ]

sec (sec tan ) sec (sec tan )

2sec tan

dy x x

dxd d

y x x xdx dxd d

x x x xdx dxx x x x x x

x x

Solution:

Example

2.6 Exponential Functions

In general, if a1 is a positive constant, the function f(x)=ax is the exponential function with base a.

If x=n is a positive integer, then an=a a … a.

If x=0, then a0=1,

If x=-n for some positive integer n, then

If x=1/n for some positive integer n, then

If x=p/q is any rational number, then

If x is an irrational number, then

1 1( )n n

na

a a

1/n na a

/ ( )q qp q p pa a a

Rules for Exponents

The Natural Exponential Function ex

The most important exponential function used for modeling natural, physical, and economic phenomena is the natural exponential function, whose base is a special number e.

The number e is irrational, and its value is 2.718281828 to nine decimal places.

The graph of y=ex has slope 1 when it crosses the y-axis.

Derivative of the Natural Exponential Function

Example. Find the derivative of y=e-x.

Solution: 2

1 0 1( ) ( )

( )

x xx x

x x

d d e ee e

dx dx e e

2.7 The Chain Rule

Example: 2Let y=sin( ). Find .

dx

dx

Solution: 2Let . sin .u x Then y u

Example

cos , 2 .dy du

so u and xdu dx

2

2

Then by the Chain Rule,

cos (2 )

cos( )(2 )

2 cos( )

dy dy du

dx du dxu x

x x

x x

“Outside-inside” Rule

It sometimes helps to think about the Chain Rule using functional notation. If y=f(g(x)), then

In words, differentiate the “outside” function f and evaluate it at the “inside” function g(x) left alone; then multiply by the derivative of the “inside” function.

'( ( )) '( )dy

f g x g xdx

Example

Example. Differentiate sin(2x+ex) with respect to x.

Solution.

Example. Differentiate e3x with respect to x.

Solution.

sin(2 ) cos(2 ) (2 )x x xdx e x e e

dx

3 3 3(3) 3x x xde e e

dx

In general, we have

For example.sin sin sin( ) (sin ) cosx x xd de e x e x

dx dx

Repeated Use of the Chain Rule

Sometimes, we have to apply the chain rule more than once to calculate a derivative.

Find [sin(tan3 )].d

xdx

2

2

cos(tan(3 )) [tan(3 )]

cos(tan(3 ))sec (3 )(3)

3cos(tan(3 ))sec (3 )

dx xdx

x x

x x

Example.

Solution. (sinu) when tan(3 )d

u xdu

[tan( )] 3d

u when u xdx

The Chain Rule with Powers of a Function

If f is a differentiable function of u and if u is a differentiable function of x, then substituting y = f(u) into the Chain Rule formula leads to the formula

( ) '( )d duf u f u

dx dx

This result is called the generalized derivative formula for f.

For example. If f(u)=un and if u is a differentiable function of x, then we canObtain the Power Chain Rule:

1n nd duu nu

dx dx

Example: 8Find ( 2)dx

dx

Solution:

Example

8 8

7

7

7

7

Let 2, then

[( 2) ] [ ]

8

8( 2) [ 2]

8( 2) (1 0)

8( 2)

u x

d dx u

dx dxdu

udx

dx x

dx

x

x

Example: Find [ tan ].d

xdx

Solution:

Example

2

2

Let tan , then

[ tan ] [ ]

1

21

[tan ]2 tan

1(sec )

2 tan

sec

2 tan

u x

d dx u

dx dxdu

dxud

xdxx

xx

x

x

Example: 3 10Find [(1 sec ) ]

dx

dx

3

3 10 10

9

3 9 3

3 9 2

3 9 3

3 9 3

Let 1 sec , then

[(1 sec ) ] [ ]

10

10(1 sec ) [1 sec ]

10(1 sec ) (3sec (sec tan ))

10(1 sec ) (3sec tan )

30(1 sec ) sec tan

u x

d dx u

dx dxdu

udx

dx xdx

x x x x

x x x

x x x

Solution:

Example

2.8 Implicit Differentiation