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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