Identify the inner and outer functions

Quiz / Weekly #4 Feedback-More Effort Needed!

-Wording of Problems (derivative, slope at a point, slope of tangent line)

-Product / Quotient Rules!!!

-Quiz I:g and II:a

-Weekly 7 , 8 , 10 The Chain Rule4.1.1The Chain RuleIf y = f (u) is a differentiable function of u and u = g(x) is a differentiable function of x, then y = f (g(x)) is a differentiable function of x and

or, equivalently,

3Identify the inner and outer functionsComposite y = f (g(x)) Inner u = g(x) Outer y = f (u)

1.

2.

3.

4.

4The General Power RuleIf , where u is a differentiable function of x and n is a rational number, thenor, equivalently,

Find the Derivative (two ways)

Find the Derivative

HomeworkChain Rule WorksheetSimplifying Chain Rule4.1.2Factoring Out the Least PowersFind the Derivative

Factoring Out the Least PowersFind the Derivative

Factoring Out the Least PowersFind the Derivative

The Chain Rule (using trig)4.1.3Find the Derivative

Find the Derivative

Find the Derivative

Trig Tangent LineFind an equation of the tangent line to the graph of at the point (, 1). Then determine all values of x in the interval (0, 2) at which the graph of f has a horizontal tangent.

Homeworkp.153/ 1-11(O) , 21-39 (O) , 59 Implicit Differentiation4.2Find dy/dx

Guidelines for Implicit DifferentiationDifferentiate both sides of the equation with respect to x.Collect all terms involving dy / dx on the left side of the equation and move all other terms to the right side of the equation.Factor dy / dx out of the left side of the equation.Solve for dy / dx.21Find the derivative

Homeworkp.162/ 1-19odd, 49, 51ExampleDetermine the slope of the tangent line to the graph of

at the point .

ExampleDetermine the slope of the tangent line to the graph of

at the point .

Finding the Second Derivative Implicitly

Finding the Second Derivative Implicitly

ExampleFind the tangent and normal line to the graph given by

at the point .

Homeworkp.162/ 21-25odd, 27-30, 31-43oddInverse Functions1.5/3.8Definition of Inverse FunctionA function g is the inverse function of the function f if for each x in the domain of g.

and

for each x in the domain of f.

Verifying Inverse FunctionsShow that the functions are inverse functions of each other.

and

The Existence of an Inverse FunctionA function has an inverse function if and only if it is one-to-one.

If f is strictly monotonic on its entire domain, then it is one-to-one and therefore has an inverse function.

Existence of an Inverse FunctionWhich of the functions has an inverse function?

Finding an InverseFind the inverse function of.

The Derivative of an Inverse FunctionLet f be a function that is differentiable on an interval I. If f has an inverse function g, then g is differentiable at any x for which . Moreover,

ExampleLet .

a) What is the value of when x = 3?

b) What is the value of when x = 3?

Homeworkp. 44/ 1-6, 7-23odd, 43p. 170/ 28, 29bcInverse Trigonometric Functions3.8The Inverse Trigonometric FunctionsFunction

40Evaluating Inverse Trigonometric FunctionsEvaluate each function.

41Solving an Equation

42Using Right Trianglesa) Given y = arcsin x, where , find cos y.

b) Given , find tan y.

43Homework3.8 Inverse Trig Review worksheet44Derivatives of Inverse Trigonometric FunctionsFind

45Derivatives of Inverse Trigonometric Functions

46Differentiating Inverse Trigonometric Functions

47Differentiate and Simplify

48Homeworkp. 170/ 1-27odd, 31abDerivatives of Exponential and Logarithmic Functions3.9Properties of LogarithmsIf a and b are positive numbers and n is rational, then1.2.3.

4.

51Expand the following logarithms

Solving EquationsSolve 7 = ex + 1

Solve ln(2x 3) = 5Derivative of the Natural Exponential Function

Examples

Derivatives for Bases Other than e

56Find the derivative of each function.

57Homeworkp.44/ 33-38p.178/ 1-13odd, 29, 30Derivative of the Natural Logarithmic Function

ExampleFind the derivative of ln (2x).

Find the Derivative1.2.3.

61Derivatives for Bases Other than e

62ExampleDifferentiate .

ExampleDifferentiate .

ExampleDifferentiate .

Find an equation of the tangent line to the graph at the given point.

Homeworkp. 178/15-27odd, 37-41odd Comparing Variable and Constants

68Derivative Involving Absolute Value

Find the derivative

Use implicit differentiation to find dy/dx.ln xy + 5x = 30

Logarithmic DifferentiationDifferentiate .

Logarithmic Differentiation

73More Logarithmic Differentiation

74Homeworkp. 179/ 31, 33-36, 43-55odd