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

Rates of Change: Derivative at a Point

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 xh

Summary

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.

'( ) ( )df x or f xdx

( )f x x

2( )f x x

Notations

'( ) ' ( ) ( )( ) [ ( )]xdy df df x y f x D f x D f xdx dx dx

'( ) | | ( ) |x a x a x ady df df a f xdx 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

Graphing the Derivative

Given a graph y=f(x), we can plot the derivative of y=f(x) by estimating the slopes on the graph of f. That is, we plot the points (x, f’(x)) in the xy-plane and connect them with a smooth curve, which represents y=f’(x).

Example: Graph the derivative of the function y=f(x) in the figure below.

What we can learn from the graph of y=f’(x)?

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 ah

0

( ) ( )limh

f b h f bh

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.

Examples

Example.

Constant Multiple Rule

11 11 10 10

2 3 32

[ ] [ ] (11 ) 11

[ ] [ ] ( 2 ) 2

d dx x x xdx dxd d x x xdx x dx

Example.

Note: ( ) ( 1 ) 1 ( )d d d duu u udx dx dx dx

Derivative Sum Rule

Example.

Derivative Product Rule

[ ( ) ( )] ( ) '( ) ( ) '( )d f x g x f x g x g x f xdx

In function notation:

Example: 3 2Find if (2 2)(6 3 ).dy y x x xdx

Solution:

Example

Derivative Quotient Rule

2

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

d f x g x f x f x g xdx g x g x

In function notation:

Example: 3 22 4Find '( ) if .

5x xy x yx

Solution:

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.

Second- and Higher-Order Derivatives

22 2

2

'''( ) ( ) '' ( )( ) [ ( )]xd y d dy dyf x y D f x D f xdx dx dx dx

A general nth order derivative can be denoted by

( ) ( 1)n

n n nn

d d yy y D ydx dx

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

2.4 The Derivative as a Rate of Change

Thus, instantaneous rates are limits of average rates.

When we say rate of change, we mean instantaneous rate of change.

Motion Along a Line: Displacement, Velocity, Speed, Acceleration, and Jerk

Suppose that an object is moving along a s-axis so that we know its position s on that line as a function of time t: s=f(t). The displacement of the object over the time interval from t to t+∆t is ∆s = f(t+ ∆t)-f(t);

The average velocity of the object over that time interval is( ) ( )

avs f t t f tvt t

Velocity

To find the body’s velocity at the exact instant t, we take the limit of the Average velocity over the interval from t to t+ ∆t as ∆t shrinks to zero. The limit is the derivative of f with respect to t.

Besides telling how fast an object is moving, its velocity tells the direction ofMotion.

The speedometer always shows speed, which is the absolute value of velocity.Speed measures the rate of progress regardless of direction

The figure blow shows the velocity v=f’(t) of a particle moving on a coordinate line., what can you say about the movement ?

Acceleration

The rate at which a body’s velocity changes is the body’s acceleration. The acceleration measures how quickly the body picks up or loses speed.A sudden change in acceleration is called a jerk.

Example

Near the surface of the earth all bodies fall with the same constant acceleration. In fact, we have

s=(1/2)gt2 ,

where s is the distance fallen and g is the acceleration due to Earth’s gravity.

With t in seconds, the value of g at sea lever is 32 ft/ sec2 or 9.8m/sec2.

Example

Example: Figure left shows the free fall of a heavy ball bearing released from rest at time t=0.(a)How many meters does the ball fall in the first 2 sec?

(b)What is its velocity, speed, and acceleration when t=2?

2.5 Derivatives of Trigonometric Functions

Example: Find if cos .dy y x xdx

Solution:

Example

Example: cosFind if .

1 sindy xydx x

Solution:

Example

Example: A body hanging from a spring is stretched down 5 units beyondIts rest position and released at time t=0 to bob up and down. Its position at any later time is s=5cos t. What are its velocity and acceleration at time t?

sin cos 1 1tan , cot , sec , csccos sin cos sinx xx x x xx x x x

Since

We have

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

Example. Find the derivative of y=e-1/x.

2.7 The Chain Rule

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

dxdx

Solution:

Example

“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.

In general, we have

For example.

Example: find derivative of |x| when x ≠ 0.

sin sin sin( ) (sin ) cosx x xd de e x e xdx 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

Example.

Solution.

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 udx 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 nudx dx

Example: 8Find ( 2)d xdx

Solution:

Example

Example: Find [ tan ].d xdx

Solution:

Example

Example: 3 10Find [(1 sec ) ]d x

dx

Solution:

Example

2.8 Implicit Differentiation

Definition. We will say that a given equation in x and y defines the function f implicitly if the graph of y = f(x) coincides with a portion of the graph of the equation.

Example:

•The equation implicitly defines functions

2 2 1x y 2 2

1 2( ) 1 and ( ) 1f x x f x x

2x y

1 2( ) and ( )f x x f x x

•The equation implicitly defines the functions

There are two methods to differentiate the functions defined implicitly by the equation.

For example: Find / if 1dy dx xy

One way is to rewrite this equation as , from which it

follows that

1yx

2

1dydx x

Two differentiable methods

With this approach we obtain [ ] [1]

[ ] [ ] 0

0

d dxydx dxd dx y y xdx dxdyx ydxdy ydx x

The other method is to differentiate both sides of the equation before solving for y in terms of x, treating y as a differentiable function of x. The method is called implicit differentiation.

Since , 1yx

2

1dydx x

Two differentiable methods

Implicit Differentiation

Example: Use implicit differentiation to find dy / dx if

Solution:

2 2 3x y x

Example

Example: Find dy / dx if 3 3 11 0y x

Solution:

Example

Lenses, tangents and Normal Lines

In the law that describes how light changes direction as it enters a lens, the important angles are the angles the light makes with the line perpendicular to the surface of the lens at the point of entry.

This line is called the normal to the surf surface at the point of entry.

The normal is the line perpendicular to the tangent of the profile curve at the point of entry.

Example

Show that the point (2, 4) lies on the curve x3+y3-9xy=0. Then find the tangent and normal to the curve there.

Derivatives of Higher Order

Find dy2 /dx2 if 2x3-3y2=8.

2.9 Inverse Functions and Their Derivatives

A function that undoes, or inverts, the effect of a function f is called the inverse of f.

Examples

Inverse Function

Note the symbol f -1 for the inverse of f is read “f inverse”. The “-1” in f -1 isnot an exponent; f -1 (x) does not mean 1/f(x).

Finding Inverses

The process of passing from f to f -1 can be summarized as a two-step process.

1. Solve the equation y=f(x) for x. This gives f formula x=f -1(y) where x is expressed as a function of y.

2. Interchange x and y, obtaining a formula y=f -1(x), where f -1 is expressed in the conventional format with x as the independent

variable and y as the dependent variable.

Examples

Find the inverse of (a) y=3x-2.(b) y=x2,x≥0.Solution:

Derivative Rule for Inverses

Derivative Rule for Inverses

Example

Let f(x)= x3-2. Find the value of df-1/dx at x=6 = f(2) without finding a formula for f -1 (x).

2.10 Logarithmic Functions

Natural Logarithm Function

Logarithms with base e and base 10 are so important in applications thatCalculators have special keys for them. logex is written as lnx log10x is written as logx

The function y=lnx is called the natural logarithm function, and y=logx isOften called the common logarithm function.

Properties of Logarithms

Properties of ax and logax

Derivative of the Natural Logarithm Function

3Find [ln( 4)]d xdx

Example:

Solution:

Note: 1[ln ] , 0d x xdx x

Since y=lnx is the inverse function of y=ex, we have

Example

Example: Find [ln | cos |]d xdx

Solution:

Furthermore, since |x|=x when x>0 and |x|= -x when x<0,

Derivatives of au

Example:

Note that [ ] ln , [ ] , [ ]x x x x u ud d d dua a a e e e edx dx dx dx

Since ax=exlna, we can find the following result.

Derivatives of logau

Example:

Note that 1lnd duudx u dx

Since logax =lnx/lna, we can find the following result.

Logarithmic Differentiation

The derivatives of positive functions given by formulas that involve products, quotients, and powers can often be found more quickly if we take the natural logarithm of both sides before differentiating. This process is called logarithmic differentiation.

Example. Find dy/dx if 2 1/2( 1)( 3) , 1

1x xy x

x

The Number e as a Limit

2.11 Inverse Trigonometric Functions

The six basic trigonometric functions are not one-to-one (their values Repeat periodically). However, we can restrict their domains to intervals on which they are one-to-one.

Six Inverse Trigonometric Functions

Since the restricted functions are now one-to-one, they have inverse, which we denoted by

These equations are read “y equals the arcsine of x” or y equals arcsin x” and so on.

Caution: The -1 in the expressions for the inverse means “inverse.” It doesNot mean reciprocal. The reciprocal of sinx is (sinx)-1=1/sinx=cscx.

1

1

1

1

1

1

sin arcsin

cos arccos

tan arctan

cot arccot

sec arcsec

csc arccsc

y x or y x

y x or y x

y x or y x

y x or y x

y x or y x

y x or y x

Derivative of y = sin-1x

Example: Find dy/dx if

Solution:

1 2sin ( )y x

Derivative of y = tan-1x

Example: Find dy/dx if

Solution:

1tan ( )xy e

Derivative of y = sec-1x

Example: Find dy/dx if

Solution:

1 3sec (4 )y x

Derivative of the other Three

There is a much easier way to find the other three inverse trigonometric Functions-arccosine, arccotantent, and arccosecant, due to the followingIdentities:

It follows easily that the derivatives of the inverse cofunctions are the negativesof the derivatives of the corresponding inverse functions.

2.13 Linearization and Differentials

In general, the tangent to y=f(x) at a point x=a, where f is differentiable, passes through the point (a, f(a)), so its point-slope equation is y=f(a)+f’(a)(x-a).

Thus this tangent line is the graph of the linear function L(x)=f(a)+f’(a)(x-a)..For as long as this line remains close to the graph of f, L(x) gives a goodapproximation to f(x).

Linearization

Example

Find the linearization of f(x)=cosx at x=π/2.

Also an important linear approximation for roots and poewrs is

(1+x)k==1+kx (x near 0; any number k).