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Section 2.3 – Product and Quotient Rules and Higher-Order Derivatives. The Product Rule. Another way to write the Rule:. The derivative of a product of functions is NOT the product of the derivatives. If f and g are both differentiable, then: - PowerPoint PPT Presentation

Section 2.3 Product and Quotient Rules and Higher-Order Derivatives

Section 2.3 Product and Quotient Rules and Higher-Order DerivativesThe Product RuleThe derivative of a product of functions is NOT the product of the derivatives.

If f and g are both differentiable, then:

In other words, the derivative of a 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.

Another way to write the Rule:

Example 1Differentiate the function:

Product RuleSum/Difference RulePower RuleSimplify

uv

vu'uv'Example 2If h(x) = xg(x) and it is known that g(3) = 5 and g'(3) =2, find h'(3).

Product RuleFind the derivative:Evaluate the derivative:

uv

uv'uv'

The Quotient RuleThe derivative of a quotient of functions is NOT the quotient of the derivatives.

If f and g are both differentiable, then:

In other words, the derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

Another way to write the Rule:

Lo-d-Hi minus Hi-d-LoExample 1Differentiate the function:

Quotient Rule

uvuv'vu'v2Example 2Find an equation of the tangent line to the curve at the point (1,).

Find the Derivative Find the derivative (slope of the tangent line) when x=1

Use the point-slope formula to find an equation

Example 3Differentiate the function:

Sum and Difference RulesConstant Multiple RulePower RuleSimplify

The Quotient Rule is long, dont forget to rewrite if possible.Rewrite to use the power rule

Try to find the Least Common DenominatorDerivative of SecantDifferentiate f(x) = sec(x).

The derivation for tangent is in the book.Rewrite as a QuotientQuotient RuleRewrite to use Trig Identities9More Derivatives of Trigonometric FunctionsWe will assume the following to be true:

Example 1Differentiate the function:

Quotient Rule

uvuv'vu'Use the Quotient Rule

Always look to simplify

Trig Law: 1 + tan2 = sec2 Example 2Differentiate the function:

Product Rule

uvuv'vu'Use the Product RuleExample 1Let

a. Find the derivative of the function.

Example 1 (Continued)Let

b. Find the derivative of the function found in (a).

Example 1 (Continued)Let

c. Find the derivative of the function found in (b).

Example 1 (Continued)Let

d. Find the derivative of the function found in (c).

Example 1 (Continued)Let

e. Find the derivative of the function found in (d).

We have just differentiated the derivative of a function. Because the derivative of a function is a function, differentiation can be applied over and over as long as the derivative is a differentiable function. First DerivativeSecond DerivativeThird DerivativeFourth Derivativenth DerivativeHigher-Order Derivatives: Notation

Notice that for derivatives of higher order than the third, the parentheses distinguish a derivative from a power. For example:.

Example 1 (Continued)Let

f. Define the derivatives from (a-e) with the correct notation.

You should note that all higher-order derivatives of a polynomial p(x) will also be polynomials, and if p has degree n, then p(n)(x) = 0 for k n+1.

Example 2If , find . Will ever equal 0?

Find the first derivative:

Find the second derivative:

Find the third derivative:

No higher-order derivative will equal 0 since the power of the function will never be 0. It decreases by one each time.Example 3Find the second derivative of .

Find the first derivative:

Find the second derivative:Graphs of a Function and its DerivativesWhat can we say about g, g', g'' for the segment of the graph of y = g(x)?

g :g' :g'' :IncreasingPositive, IncreasingPositive As the graph increases, the tangent lines are getting steeper.Since the first derivative is increasing, the second derivative must be positive.Graphs of a Function and its Derivatives

What can we say about g, g', g'' for the segment of the graph of y = g(x)?g :g' :g'' :DecreasingNegative, DecreasingNegativeAs the graph decreases, the tangent lines are getting less steep.Since the first derivative is decreasing, the second derivative must be negative.Graphs of a Function and its Derivatives

What can we say about g, g', g'' for the segment of the graph of y = g(x)?g :g' :g'' :DecreasingNegative, IncreasingPositiveAs the graph decreases, the tangent lines are steeper.Since the first derivative is increasing, the second derivative must be positive.Graphs of a Function and its DerivativesWhat can we say about g, g', g'' for the segment of the graph of y = g(x)?

g :g' :g'' :IncreasingPositive, DecreasingNegativeAs the graph increases, the tangent lines are getting less steep.Since the first derivative is decreasing, the second derivative must be negative.

Graphs of a Function and its DerivativesWhat can we say about g, g', g'' for the segment of the graph of y = g(x)?g :g' :g'' :DecreasingNegative, DecreasingNegativeFind the pieces of this graph that compare to the previous graphs.On the left side :g :g' :g'' :DecreasingNegative, IncreasingPositiveOn the right side :Average AccelerationExample: Estimate the velocity at time 5 for graph of velocity at time t below.1234562-2-4v(t)tAcceleration is the rate at which an object changes its velocity. An object is accelerating if it is changing its velocity.Find the average rate of change of velocity for times that are close and enclose time 5.

Instantaneous AccelerationIf s = s(t) is the position function of an object that moves in a straight line, we know that its first derivative represents the velocity v(t) of the object as a function of time.

The instantaneous rate of change of velocity with respect to time is called the acceleration a(t) of an object. Thus, the acceleration function is the derivative of the velocity function and is therefore the second derivative of the position function.

Position, Velocity, and AccelerationPosition, Velocity, and Acceleration are related in the following manner:

Position:

Velocity:

Acceleration:

Units = Measure of length (ft, m, km, etc)The object isMoving right/up when v(t) > 0Moving left/down when v(t) < 0Still or changing directions when v(t) = 0Units = Distance/Time (mph, m/s, ft/hr, etc)Speed = absolute value of v(t)

Units = (Distance/Time)/Time (m/s2)Example1234562-2-4v(t)tExample: The graph below at left is a graph of a particles velocity at time t. Graph the objects acceleration where it exists and answer the questions below1234562-2-4a(t)tm = 2Cornerm = 0m = -4m = 4When is the particle speeding up?

When is the particle traveling at a constant speed?

When is the function slowing down?Positive acceleration and positive velocity(0,2)Negative acceleration and Negative velocityU (5,6)(2,4)0 accelerationAnd constant velocityNegative acceleration and Positive velocity(4,5)U (6,7)Positive acceleration and Negative velocityMoving away from x-axis. HorizontalMoving towards the x-axis.Speeding Up and Slowing DownAn object is SPEEDING UP when the following occur:Algebraic: If the velocity and the acceleration agree in signGraphical: If the velocitycurve is moving AWAY from the x-axis

An object is traveling at a CONSTANT SPEED when the following occur:Algebraic: Velocity is constant and acceleration is 0.Graphically: The velocity curve is horizontal

An object is SLOWING DOWN when the following occur:Algebraic: Velocity and acceleration disagree in signGraphically: The velocity curve is moving towards the x-axisExample 3The position of a particle is given by the equation

where t is measured in seconds and s in meters. (a) Find the acceleration at time t.

The derivative of the position function is the velocity function.

The derivative of the velocity function is the acceleration function.Example 3 (continued)The position of a particle is given by the equation

where t is measured in seconds and s in meters. (b) What is the acceleration after 4 seconds?

(c) Is the particle speeding up, slowing down, or traveling at a constant speed at 4 seconds?

m/s2

m/s

m/s2Since the velocity and acceleration agree in signs, the particle is speeding up.

Example 3 (continued)The position of a particle is given by the equation

where t is measured in seconds and s in meters. (d) When is the particle speeding up? When is it slowing down?

Velocity:Acceleration:+++Speeding Up:(1,2)U (3,)Slowing Down:(0,1)U (2,3)