Chapter 11 Differentiation

Chapter 11: Differentiation Chapter Objectives To compute derivatives by using the limit definition. To develop basic differentiation rules. To interpret the derivative as an instantaneous rate of change. To apply the product and quotient rules. To apply the chain rule.

Chapter Outline The Derivative Rules for Differentiation Chapter 11: Differentiation Chapter Outline The Derivative Rules for Differentiation The Derivative as a Rate of Change The Product Rule and the Quotient Rule The Chain Rule and the Power Rule 11.1) 11.2) 11.3) 11.4) 11.5)

11.1 The Derivative One of the 1st problems of differential calculus is the tangent line problem Note that some curves have no tangent line: Applications to Business and Economics: Marginal Cost, Marginal Profit, Minimizing Cost, Maximizing profit, Etc…

Therefore the equation of tangent line at (1,1) is:

Leibnitz notation Notation:

Let’s construct equation of tangent line at a different point:

FYI

p q

a. For f(x) = x2, it must be continuous for all x. Foregoing a rigorous proof, the idea is that if a function is smooth enough to have a tangent line, so it must be continuous without jumps etc… a. For f(x) = x2, it must be continuous for all x. b. For f(p) =(1/2)p, it is not continuous at p = 0, thus the derivative does not exist at p = 0. Note that converse is not necessarily true as for f(x)=|x| I.e. this function is continuous at x = 0, but NOT differentiable

Section 11.1 HW 7 25

Section 11.1 HW 7 25

11.2 Rules for Differentiation Example:

This rule is actually valid for any real power!

Differentiate the following functions: Solution: a. Chapter 11: Differentiation 11.2 Rules for Differentiation Example 3 – Rewriting Functions in the Form xn Differentiate the following functions: Solution: a. b.

Example – Using the Power Rule In Example 2(c), note that before differentiating, 1/x2 was rewritten as x-2. Rewriting is the first step in many differentiation problems.

Differentiate the following functions: Chapter 11: Differentiation 11.2 Rules for Differentiation Example 5 – Differentiating Sums and Differences of Functions Differentiate the following functions:

Chapter 11: Differentiation 11.2 Rules for Differentiation Example 5 – Differentiating Sums and Differences of Functions

Find an equation of the tangent line to the curve when x = 1. Chapter 11: Differentiation 11.2 Rules for Differentiation Example 7 – Finding an Equation of a Tangent Line Find an equation of the tangent line to the curve when x = 1. Solution: The slope equation is When x = 1, The equation is

Section 11.2 HW 67 69-71 83 87

11.3 The Derivative as a Rate of Change

Average velocity is given by Velocity at time t is given by Chapter 11: Differentiation Example 1 – Finding Average Velocity and Velocity Average velocity is given by Velocity at time t is given by Suppose the position function of an object moving along a number line is given by s = f(t) = 3t2 + 5, where t is in seconds and s is in meters. Find the average velocity over the interval [10, 10.1]. b. Find the velocity when t = 10.

b. Velocity at time t is given by Chapter 11: Differentiation 11.3 The Derivative as a Rate of Change Example 1 – Finding Average Velocity and Velocity Solution: a. When t = 10, b. Velocity at time t is given by When t = 10, the velocity is s = f(t) = 3t2 + 5

Generalization:

Demand and supply curves were discussed in Ch3. They give dependence between the price/per unit and the quantity demanded by consumers of quantity produced by suppliers

Solution: Rate of change of V with respect to r is Chapter 11: Differentiation 11.3 The Derivative as a Rate of Change Example 5 – Rate of Change of Volume A spherical balloon is being filled with air. Find the rate of change of the volume of air in the balloon with respect to its radius. Evaluate this rate of change when the radius is 2 ft. Solution: Rate of change of V with respect to r is When r = 2 ft, This means that when radius is 2, changing it by 1 unit, will change volume by 16π units.

In general:

Determine the relative and percentage rates of change of Chapter 11: Differentiation 11.3 The Derivative as a Rate of Change Example 9 – Relative and Percentage Rates of Change Determine the relative and percentage rates of change of when x = 5. Solution:

11.3 HW 7 9 25 27 35 39 45

11.4 The Product Rule and the Quotient Rule Alternative method: -- seems easier, but for products of transcendental functions (exp, ln, sin/cos etc…) it is essential

Application to tangent lines:

The Quotient Rule

Note that in the demand equation: p = price/unit q = number of units

11.4 HW 35 37 50-53 65 69 Differentiate:

11.4 HW 35 37 50-53 65 69

Generalizes to:

HW 13 23 27 29 39 47 69 80

HW 13 23 27 29 39 47 69 80