Chapter 3 The DerivativeChapter 3 The Derivative

Definition, Interpretations, and Definition, Interpretations, and RulesRules

Average Rate of ChangeAverage Rate of Change

For y = f(x), the For y = f(x), the average rate of change average rate of change from x = a to x = a+hfrom x = a to x = a+h is is

f(a + h) - f(a)

h

Average Rate of Change, cont.Average Rate of Change, cont.

Graphically, the average rate of change can Graphically, the average rate of change can be interpreted asbe interpreted as

the slope of the secant line to the graph the slope of the secant line to the graph through the points (a, f(a)) and (a+h, f(a+h)).through the points (a, f(a)) and (a+h, f(a+h)).

Instantaneous Rate of ChangeInstantaneous Rate of Change

If y = f(x), the If y = f(x), the instantaneous rate of change instantaneous rate of change at x = aat x = a is is

h 0

f(a + h) - f(a)lim

h

The DerivativeThe Derivative

For y = f(x), we define the derivative of f at x, For y = f(x), we define the derivative of f at x, denoted f’(x), to bedenoted f’(x), to be

0

( ) ( )'( ) limh

f x h f xf x

h

Interpretations of the DerivativeInterpretations of the Derivative

The derivative of a function f is a new The derivative of a function f is a new function f’. The derivative has various function f’. The derivative has various applications and interpretations, including:applications and interpretations, including:

1. Slope of the Tangent Line to the graph of 1. Slope of the Tangent Line to the graph of f at the point (x, f(x)).f at the point (x, f(x)).

2. Slope of the graph of f at the point (x, 2. Slope of the graph of f at the point (x, f(x))f(x))

3. Instantaneous Rate of Change of y = f(x) 3. Instantaneous Rate of Change of y = f(x) with respect to x.with respect to x.

DifferentiationDifferentiation

The process of finding the derivative of a The process of finding the derivative of a function is calledfunction is called

differentiation.differentiation. That is, the derivative of a function is That is, the derivative of a function is

obtained byobtained by differentiating the function.differentiating the function.

Nonexistence of the DerivativeNonexistence of the Derivative

The existence of a derivative at x = a The existence of a derivative at x = a depends on the existence of a limit at x = a, depends on the existence of a limit at x = a, that is, on the existence of that is, on the existence of

0

( ) ( )'( ) limh

f a h f af a

h

Nonexistence, cont.Nonexistence, cont.

So, if the limit does not exist at a point x = a, So, if the limit does not exist at a point x = a, we say that the function f iswe say that the function f is

nondifferentiable at x = a, or f’(a) does not nondifferentiable at x = a, or f’(a) does not exist.exist.

Graphically, this means if there is a break in Graphically, this means if there is a break in the graph at a point, then the derivative the graph at a point, then the derivative does not exist at that point.does not exist at that point.

Nonexistence, cont.Nonexistence, cont.

There are other ways to recognize the There are other ways to recognize the points on the graph of f where f’(a) does not points on the graph of f where f’(a) does not exist. They areexist. They are

1. The graph of f has a hole at x = a.1. The graph of f has a hole at x = a. 2. The graph of f has a sharp corner at x = 2. The graph of f has a sharp corner at x =

a.a. 3. The graph of f has a vertical tangent line 3. The graph of f has a vertical tangent line

at x = a.at x = a.

Finding or approximating f’(x).Finding or approximating f’(x).

We have seen three different ways to find or We have seen three different ways to find or apoproximate f’(x). They are;apoproximate f’(x). They are;

1. Numerically, by computing the difference quotient for 1. Numerically, by computing the difference quotient for small values of x.small values of x.

2. Graphically, by estimating the slope of a tangent line 2. Graphically, by estimating the slope of a tangent line at the point (x, f(x)).at the point (x, f(x)).

3. Algebraically, by using the two-step limiting process 3. Algebraically, by using the two-step limiting process to evaluateto evaluate

0

( ) ( )'( ) limh

f a h f af a

h

Derivative NotationDerivative Notation

Given y = f(x), we can represent the Given y = f(x), we can represent the derivative of f at x in three ways;derivative of f at x in three ways;

1. f’(x)1. f’(x) 2. y’2. y’ 3.dy/dx3.dy/dx

Derivative RulesDerivative Rules

Derivative of a Constant Function RuleDerivative of a Constant Function Rule If y = f(x) = C, thenIf y = f(x) = C, then f’(x) =0f’(x) =0 In words, the rule can be stated;In words, the rule can be stated; The derivative of any constant function is 0.The derivative of any constant function is 0.

Derivative Rules, cont.Derivative Rules, cont.

Power RulePower Rule

n

n-1

y = f(x) = x , where n is any real number, then

f '(x) = nx

If

Rules, cont.Rules, cont.

Constant Times a Function RuleConstant Times a Function Rule If y = f(x) = ku(x), thenIf y = f(x) = ku(x), then f ‘(x) = ku’(x)f ‘(x) = ku’(x) In words, the rule can be stated;In words, the rule can be stated; The derivative of a constant times a The derivative of a constant times a

differentiable function is the constant times differentiable function is the constant times the derivative of the function.the derivative of the function.

Rules, cont.Rules, cont.

Sum and Difference RuleSum and Difference Rule

If y = f(x) = u(x) v(x), then

f '(x) = u '(x) v '(x)In words, the rule can be stated;

The derivative of the sum or difference

of two functions is the sum or difference

of the derivatives.

Rules, cont.Rules, cont.

Product RuleProduct Rule

If y = f(x) = F(x) S(x), then

f '(x) = F(x) S '(x) + S(x) F '(x)

In words, the derivative of the product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.

Rules, cont.Rules, cont.

Quotient RuleQuotient Rule

T(x)If f(x) = , then

B(x)

2

B(x) T '(x) - T(x) B '(x) f '(x) =

[B(x)]

In words, the derivative of the quotient of two functions is the bottom function times the derivative of the top function minus the top function times the derivative of the bottom function, all divided by the bottom function squared.

Rules, cont.Rules, cont.

General Power RuleGeneral Power Rule

If u(x) is a differentiable function,

n is any real number, and

ny = f(x) = [u(x)] , then

n-1f '(x) = n[u(x)] u ' (x)

LimitsLimits

We writeWe write

x clim f(x) = L

If the functional value f(x) is close to the single real number L whenever x is close to, but not equal to, c.

Limits, cont.Limits, cont.

We write We write

-x clim f(x) = K

And call K the limit from the left if f(x) is close to K whenever x is close to c, but to the left of c.

Limits, cont.Limits, cont.

And we writeAnd we write

+x clim f(x) = L

And call L the limit from the right if f(x) is close to L whenever x is close to c, but to the right of c.

On the Existence of a LimitOn the Existence of a Limit

In order for a limit to exist,In order for a limit to exist, the limit from the left and the limit from the the limit from the left and the limit from the

rightright must exist and be equal.must exist and be equal.

Properties of LimitsProperties of Limits

Let Let ff and and gg be two functions, and assume that, be two functions, and assume that,

x c x clim f(x) = L lim g(x) = M then

x c x c x c1. lim [f(x) g(x)] = lim f(x) lim g(x) = L M

x c x c2. lim k f(x) = k lim f(x) = k L

x c x c x c3. lim [f(x) g(x)] = [lim f(x)][lim g(x)] = LM

x cx c

x c

[lim f(x)]f(x) L4. lim = =

g(x) [lim g(x)] M