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7th Edition

Ron Larson + 1 other

Publisher: Cengage Learning

ISBN: 9781337552516

Chapter 3.5, Problem 3E

Textbook Problem

Chain Rule How is the Chain Rule applied when finding

Expert Solution

To determine

The way to apply chain rule when finding

The chain rule is a way to differentiate functions within functions.

The implicit differentiation is a way to take the derivative of a term with respect to another variable without having to isolate either variable.

Consider the following example:

Define the inner function and the outer function,

Differentiate the outer function via power rule and chain rule:

Calculus: Early Transcendental Functions

Show all chapter solutions

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Using the Product Rule In Exercises 5-10, use the...Ch. 3.3 - Using the Product Rule In Exercises 5-10, use the...Ch. 3.3 - Using the Product Rule In Exercises 5-10, use the...Ch. 3.3 - Using the Product Rule In Exercises 5-10, use the...Ch. 3.3 - Using the Quotient Rule In Exercises 11-16, use...Ch. 3.3 - Using the Quotient Rule In Exercises 11-16, use...Ch. 3.3 - Using the Quotient Rule In Exercises 11-16, use...Ch. 3.3 - Using the Quotient Rule In Exercises 11-16, use...Ch. 3.3 - Using the Quotient Rule In Exercises 11-16, use...Ch. 3.3 - Using the Quotient Rule In Exercises 11-16, use...Ch. 3.3 - Finding and Evaluating a Derivative In Exercises...Ch. 3.3 - Finding and Evaluating a Derivative In Exercises...Ch. 3.3 - Finding and Evaluating a Derivative In Exercises...Ch. 3.3 - Finding and Evaluating a Derivative In Exercises...Ch. 3.3 - Finding and valuating a Derivative In Exercises...Ch. 3.3 - Finding and Evaluating a Derivative In Exercises...Ch. 3.3 - Finding and Evaluating a Derivative In Exercises...Ch. 3.3 - Finding and Evaluating a Derivative In Exercises...Ch. 3.3 - Using the Constant Multiple Rule In Exercises...Ch. 3.3 - Using the Constant Multiple Rule In Exercises...Ch. 3.3 - Using the Constant Multiple Rule In Exercises...Ch. 3.3 - Using the Constant Multiple Rule In Exercises...Ch. 3.3 - Using the Constant Multiple Rule In Exercises...Ch. 3.3 - Using the Constant Multiple Rule In Exercises...Ch. 3.3 - Finding a Derivative In Exercises 31-42 find the...Ch. 3.3 - Finding a Derivative In Exercises 31-42, find the...Ch. 3.3 - Finding a Derivative In Exercises 31-42, find the...Ch. 3.3 - Finding a Derivative In Exercises 31-42, find the...Ch. 3.3 - Finding a Derivative In Exercises 31-42, find the...Ch. 3.3 - Finding a Derivative In Exercises 31-42, find the...Ch. 3.3 - Finding a Derivative In Exercises 31-42, find the...Ch. 3.3 - Finding a Derivative In Exercises 31-42, find the...Ch. 3.3 - Finding a Derivative In Exercises 31-42, find the...Ch. 3.3 - Finding a Derivative In Exercises 31-42, find the...Ch. 3.3 - Finding a Derivative In Exercises 31-42, find the...Ch. 3.3 - Finding a Derivative In Exercises 31-42, find the...Ch. 3.3 - Finding a Derivative of a Transcendental Function...Ch. 3.3 - Finding a Derivative of a Transcendental Function...Ch. 3.3 - Finding a Derivative of a Transcendental Function...Ch. 3.3 - Finding a Derivative of a Transcendental Function...Ch. 3.3 - Finding a Derivative of a Transcendental Function...Ch. 3.3 - Finding a Derivative of a Transcendental Function...Ch. 3.3 - Finding a Derivative of a Transcendental Function...Ch. 3.3 - Finding a Derivative of a Transcendental Function...Ch. 3.3 - Finding a Derivative of a Transcendental Function...Ch. 3.3 - Finding a Derivative of a Transcendental Function...Ch. 3.3 - Finding a Derivative of a Transcendental Function...Ch. 3.3 - Finding a Derivative of a Transcendental Function...Ch. 3.3 - Finding a Derivative of a Transcendental Function...Ch. 3.3 - Finding a Derivative of a Transcendental Function...Ch. 3.3 - Finding a Derivative of a Transcendental Function...Ch. 3.3 - Finding a Derivative of a Transcendental Function...Ch. 3.3 - Finding a Derivative of a Transcendental Function...Ch. 3.3 - Finding a Derivative of a Transcendental Function...Ch. 3.3 - Finding a Derivative Using Technology In Exercises...Ch. 3.3 - Finding a Derivative Using Technology In Exercises...Ch. 3.3 - Finding the Slope of a Graph In Exercises 63-66,...Ch. 3.3 - Finding the Slope of a Graph In Exercises 63-66....Ch. 3.3 - Finding the Slope of a Graph In Exercises 63-66,...Ch. 3.3 - Finding the Slope of a Graph In Exercises 63-66,...Ch. 3.3 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.3 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.3 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.3 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.3 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.3 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.3 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.3 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.3 - Famous Curves In Exercises 75-78, find an equation...Ch. 3.3 - Famous Curves In Exercises 75-78, find an equation...Ch. 3.3 - Famous Curves In Exercises 75-78, find an equation...Ch. 3.3 - Famous Curves In Exercises 75-78, find an equation...Ch. 3.3 - Horizontal Tangent Line In Exercises 79-82,...Ch. 3.3 - Horizontal Tangent Line In Exercises...Ch. 3.3 - Horizontal Tangent Line In Exercises...Ch. 3.3 - Horizontal Tangent Line In Exercises 79-82,...Ch. 3.3 - Tangent Lines Find equations of the tangent lines...Ch. 3.3 - Tangent Lines Find equations of the tangent lines...Ch. 3.3 - Exploring a Relationship In Exercises 85 and 86,...Ch. 3.3 - Exploring a Relationship In Exercises 85 and 86,...Ch. 3.3 - Finding Derivatives In Exercises 87 and 88, use...Ch. 3.3 - Finding Derivatives In Exercises 87 and 88, use...Ch. 3.3 - Area The length of a rectangle is given by 6t+5...Ch. 3.3 - Volume The radius of a right circular cylinder is...Ch. 3.3 - Inventory Replenishment The ordering and...Ch. 3.3 - 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(a)...Ch. 3.3 - Rate of Change Determine whether there exist any...Ch. 3.3 - Modeling Data The table shows the health care...Ch. 3.3 - Satellites When satellites observe Earth, they can...Ch. 3.3 - Finding a Second Derivative In Exercises 97-108,...Ch. 3.3 - Finding a Second Derivative In Exercises 97-108,...Ch. 3.3 - Finding a Second Derivative In Exercises 97-108,...Ch. 3.3 - Finding a Second Derivative In Exercises 97-108,...Ch. 3.3 - Finding a Second Derivative In Exercises 97-108,...Ch. 3.3 - Finding a Second Derivative In Exercises 97-108,...Ch. 3.3 - Finding a Second Derivative In Exercises 97-108,...Ch. 3.3 - Finding a Second Derivative In Exercises 97-108,...Ch. 3.3 - Finding a Second Derivative In Exercises 97-108,...Ch. 3.3 - Finding a Second Derivative In Exercises 97-108,...Ch. 3.3 - Finding a Second Derivative In Exercises 97-108,...Ch. 3.3 - Finding a Second Derivative In Exercises 97-108,...Ch. 3.3 - Finding a Higher-Order Derivative In Exercises...Ch. 3.3 - Finding a Higher-Order Derivative In Exercises...Ch. 3.3 - Finding a Higher-Order Derivative In Exercises...Ch. 3.3 - Finding a Higher-Order Derivative In Exercises...Ch. 3.3 - Using Relationships In Exercises 113-116, use the...Ch. 3.3 - Using Relationships In Exercises 113-116, use the...Ch. 3.3 - Using Relationships In Exercises 113-116, use the...Ch. 3.3 - Using Relationships In Exercises 113-116, use the...Ch. 3.3 - Higher-Order Derivatives Polynomials of what...Ch. 3.3 - Differentiation of Piecewise Functions Describe...Ch. 3.3 - Identifying Graphs In Exercises 119 and 120, the...Ch. 3.3 - Identify Graphs In Exercises 119 and 120, the...Ch. 3.3 - Sketching Graphs In Exercises 121 and 122, the...Ch. 3.3 - Sketching Graphs In Exercises 121 and 122, the...Ch. 3.3 - Sketching a Graph Sketch the graph of a...Ch. 3.3 - Sketching a Graph Sketch the graph of a...Ch. 3.3 - Acceleration The velocity of an object is...Ch. 3.3 - Acceleration The velocity of an automobile...Ch. 3.3 - Stopping Distance A car is traveling at a rate of...Ch. 3.3 - HOW DO YOU SEE it? The figure shows the graphs of...Ch. 3.3 - Finding a Pattern In Exercises 129 and 130,...Ch. 3.3 - Finding a Pattern In Exercises 129 and 130,...Ch. 3.3 - Finding a Pattern Consider the function...Ch. 3.3 - Finding a Pattern Develop a general rule for the...Ch. 3.3 - Finding a Pattern In Exercises 133 and 134, find...Ch. 3.3 - Finding a Pattern In Exercises 133 and 134, find...Ch. 3.3 - Differential Equations In Exercises 135-138,...Ch. 3.3 - Differential equations In Exercises 135-138,...Ch. 3.3 - Differential Equations In Exercises 135-138,...Ch. 3.3 - Differential Equations In Exercises 135-138,...Ch. 3.3 - True or False? In Exercises 139-144, determine...Ch. 3.3 - True or False? In Exercises 139-144, determine...Ch. 3.3 - True or False? In Exercises 139-144, determine...Ch. 3.3 - True or False? In Exercises 139-144, determine...Ch. 3.3 - True or False? In Exercises 139-144, determine...Ch. 3.3 - True or False? In Exercises 139-144, determine...Ch. 3.3 - Proof Use the Product Rule twice to prove that if...Ch. 3.3 - Think About It Let f and p be functions whose...Ch. 3.4 - Chain Rule Describe the Chain Rule for the...Ch. 3.4 - General Power Rule What is the difference between...Ch. 3.4 - Natural Logarithmic Functions Explain why the...Ch. 3.4 - Derivatives for Bases Other than e What are the...Ch. 3.4 - Decomposition of a Composite Function In Exercises...Ch. 3.4 - Decomposition of a Composite Function In Exercises...Ch. 3.4 - Decomposition of a Composite Function In Exercises...Ch. 3.4 - Decomposition of a Composite Function In Exercises...Ch. 3.4 - Decomposition of a Composite Function In Exercises...Ch. 3.4 - Decomposition of a Composite Function In Exercises...Ch. 3.4 - Decomposition of a Composite Function In Exercises...Ch. 3.4 - Decomposition of a Composite Function In Exercises...Ch. 3.4 - Finding a Derivative In Exercises 13-32, find the...Ch. 3.4 - Finding a Derivative In Exercises 13-32, find the...Ch. 3.4 - Finding a Derivative In Exercises 13-32, find the...Ch. 3.4 - Finding a Derivative In Exercises 13-32, find the...Ch. 3.4 - Finding a Derivative In Exercises 13-32, find the...Ch. 3.4 - Finding a Derivative In Exercises 13-32, find the...Ch. 3.4 - Finding a Derivative In Exercises 13-32, find the...Ch. 3.4 - Finding a Derivative In Exercises 13-32, find the...Ch. 3.4 - Finding a Derivative In Exercises 13-32, find the...Ch. 3.4 - Finding a Derivative In Exercises 13-32, find the...Ch. 3.4 - Finding a Derivative In Exercises 13-32, find the...Ch. 3.4 - Finding a Derivative In Exercises 13-32, find the...Ch. 3.4 - Finding a Derivative In Exercises 13-32, find the...Ch. 3.4 - Finding a Derivative In Exercises 13-32, find the...Ch. 3.4 - Finding a Derivative In Exercises 13-32, find the...Ch. 3.4 - Finding a Derivative In Exercises 13-32, find the...Ch. 3.4 - Finding a Derivative In Exercises 13-32, find the...Ch. 3.4 - Finding a Derivative In Exercises 13-32, find the...Ch. 3.4 - Finding a Derivative In Exercises 13-32, find the...Ch. 3.4 - Finding a Derivative In Exercises 13-32, find the...Ch. 3.4 - Finding a Derivative of a Trigonometric Function...Ch. 3.4 - Finding a Derivative of a Trigonometric Function...Ch. 3.4 - Finding a Derivative of a Trigonometric Function...Ch. 3.4 - Finding a Derivative of a Trigonometric Function...Ch. 3.4 - Finding a Derivative of a Trigonometric Function...Ch. 3.4 - Finding a Derivative of a Trigonometric Function...Ch. 3.4 - Finding a Derivative of a Trigonometric Function...Ch. 3.4 - Finding a Derivative of a Trigonometric Function...Ch. 3.4 - Finding a Derivative of a Trigonometric Function...Ch. 3.4 - Finding a Derivative of a Trigonometric Function...Ch. 3.4 - Finding a Derivative of a Trigonometric Function...Ch. 3.4 - Finding a Derivative of a Trigonometric Function...Ch. 3.4 - Finding a Derivative of a Trigonometric Function...Ch. 3.4 - Finding a Derivative of a Trigonometric Function...Ch. 3.4 - Finding a Derivative of a Trigonometric Function...Ch. 3.4 - Finding a Derivative of a Trigonometric Function...Ch. 3.4 - Finding a Derivative of a Trigonometric Function...Ch. 3.4 - Finding a Derivative of a Trigonometric Function...Ch. 3.4 - Finding a Derivative Using Technology In Exercises...Ch. 3.4 - Finding a Derivative Using Technology In Exercises...Ch. 3.4 - Finding a Derivative Using Technology In Exercises...Ch. 3.4 - Finding a Derivative Using Technology In Exercises...Ch. 3.4 - Finding a Derivative Using Technology In Exercises...Ch. 3.4 - Finding a Derivative Using Technology In Exercises...Ch. 3.4 - Finding a Derivative of an Exponential Function In...Ch. 3.4 - Finding a Derivative of an Exponential Function In...Ch. 3.4 - Finding a Derivative of an Exponential Function In...Ch. 3.4 - Finding a Derivative of an Exponential Function In...Ch. 3.4 - Finding a Derivative of an Exponential Function In...Ch. 3.4 - Finding a Derivative of an Exponential Function In...Ch. 3.4 - Finding a Derivative of an Exponential Function In...Ch. 3.4 - Finding a Derivative of an Exponential Function In...Ch. 3.4 - Finding a Derivative of an Exponential Function In...Ch. 3.4 - Finding a Derivative of an Exponential Function In...Ch. 3.4 - Finding a Derivative of an Exponential Function In...Ch. 3.4 - Finding a Derivative of an Exponential Function In...Ch. 3.4 - Finding a Derivative of an Exponential Function In...Ch. 3.4 - Finding a Derivative of an Exponential Function In...Ch. 3.4 - Finding a Derivative of an Exponential Function In...Ch. 3.4 - Finding a Derivative of an Exponential Function In...Ch. 3.4 - Finding a Derivative of a Logarithmic Function In...Ch. 3.4 - Finding a Derivative of a Logarithmic Function In...Ch. 3.4 - Finding a Derivative of a Logarithmic Function In...Ch. 3.4 - Finding a Derivative of a Logarithmic Function In...Ch. 3.4 - Finding a Derivative of a Logarithmic Function In...Ch. 3.4 - Finding a Derivative of a Logarithmic Function In...Ch. 3.4 - Finding a Derivative of a Logarithmic Function In...Ch. 3.4 - Finding a Derivative of a Logarithmic Function In...Ch. 3.4 - Finding a Derivative of a Logarithmic Function In...Ch. 3.4 - Finding a Derivative of a Logarithmic Function In...Ch. 3.4 - Finding a Derivative of a Logarithmic Function In...Ch. 3.4 - Finding a Derivative of a Logarithmic Function In...Ch. 3.4 - Finding a Derivative of a Logarithmic Function In...Ch. 3.4 - Finding a Derivative of a Logarithmic Function In...Ch. 3.4 - Finding a Derivative of a Logarithmic Function In...Ch. 3.4 - Finding a Derivative of a Logarithmic Function In...Ch. 3.4 - Finding a Derivative of a Logarithmic Function In...Ch. 3.4 - Finding a Derivative of a Logarithmic Function In...Ch. 3.4 - Finding a Derivative of a Logarithmic Function In...Ch. 3.4 - Finding a Derivative of a Logarithmic Function In...Ch. 3.4 - Finding a Derivative of a Logarithmic Function In...Ch. 3.4 - Finding a Derivative of a Logarithmic Function In...Ch. 3.4 - Slope of a Tangent Line In Exercises 95 and 96....Ch. 3.4 - Slope of a Tangent Line In Exercises 95 and 96....Ch. 3.4 - Slope of a Tangent Line In Exercises 97-100. find...Ch. 3.4 - Slope of a Tangent Line In Exercises 97-100, find...Ch. 3.4 - Slope of a Tangent Line In Exercises 97-100, find...Ch. 3.4 - Slope of a Tangent Line In Exercises 97-100, find...Ch. 3.4 - Finding the Slope of a Graph In Exercises 101-108,...Ch. 3.4 - Finding the Slope of a Graph In Exercises 101-108,...Ch. 3.4 - Finding the Slope of a Graph In Exercises 101-108,...Ch. 3.4 - Finding the Slope of a Graph In Exercises 101-108,...Ch. 3.4 - Finding the Slope of a Graph In Exercises 101-108,...Ch. 3.4 - Finding the Slope of a Graph In Exercises 101-108,...Ch. 3.4 - Finding the Slope of a Graph In Exercises 101-108,...Ch. 3.4 - Finding the Slope of a Graph In Exercises 101-108,...Ch. 3.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.4 - Famous Curves In Exercises 117 and 118, find an...Ch. 3.4 - Famous Curves In Exercises 117 and 118, find an...Ch. 3.4 - Horizontal Tangent Line Determine the point(s) in...Ch. 3.4 - Horizontal Tangent Line Determine the point(s) at...Ch. 3.4 - Finding a Second Derivative In Exercises 121-128,...Ch. 3.4 - Finding a Second Derivative In Exercises 121-128,...Ch. 3.4 - Finding a Second Derivative In Exercises 121-128,...Ch. 3.4 - Finding a Second Derivative In Exercises 121-128,...Ch. 3.4 - Finding a Second Derivative In Exercises 121-128,...Ch. 3.4 - Finding a Second Derivative In Exercises 121-128,...Ch. 3.4 - Finding a Second Derivative In Exercises 121-128,...Ch. 3.4 - Finding a Second Derivative In Exercises 121-128,...Ch. 3.4 - Evaluating a Second Derivative In Exercises...Ch. 3.4 - Evaluating a Second Derivative In Exercises...Ch. 3.4 - Evaluating a Second Derivative In Exercises...Ch. 3.4 - Evaluating a Second Derivative In Exercises...Ch. 3.4 - Finding a Derivative In Exercises 133-148, find...Ch. 3.4 - Finding a Derivative In Exercises 133-148, find...Ch. 3.4 - Finding a Derivative In Exercises 133-148, find...Ch. 3.4 - Finding a Derivative In Exercises 133-148, find...Ch. 3.4 - Finding a Derivative In Exercises 133-148, find...Ch. 3.4 - Finding a Derivative In Exercises 133-148, find...Ch. 3.4 - Finding a Derivative In Exercises 133-148, find...Ch. 3.4 - Finding a Derivative In Exercises 133-148, find...Ch. 3.4 - Finding a Derivative In Exercises 133-148, find...Ch. 3.4 - Finding a Derivative In Exercises 133-148, find...Ch. 3.4 - Finding a Derivative In Exercises 133-148, find...Ch. 3.4 - Finding a Derivative In Exercises 133-148, find...Ch. 3.4 - Finding a Derivative In Exercises 133-148, find...Ch. 3.4 - Finding a Derivative In Exercises 133-148, find...Ch. 3.4 - Finding a Derivative In Exercises 133-148, find...Ch. 3.4 - Finding a Derivative In Exercises 133-148, find...Ch. 3.4 - Identifying Graphs In Exercises 149 and 150, the...Ch. 3.4 - Identifying Graphs In Exercises 149 and 150, the...Ch. 3.4 - Describing Relationships Describe the relationship...Ch. 3.4 - Comparing Methods Consider the function...Ch. 3.4 - Think About It The table shows some values of the...Ch. 3.4 - Using Relationships Given that g(5)=3, g(5)=6,...Ch. 3.4 - Finding Derivatives In Exercises 155 and 156, the...Ch. 3.4 - Finding Derivatives In Exercises 155 and 156, the...Ch. 3.4 - Pendulum A 15-centimeter pendulum moves according...Ch. 3.4 - Harmonic Motion The displacement from equilibrium...Ch. 3.4 - Doppler Effect The frequency F of a fire truck...Ch. 3.4 - HOW DO YOU SEE IT? The cost C (in dollars) of...Ch. 3.4 - Inflation When the annual rate of inflation...Ch. 3.4 - Wave Motion A buoy oscillates in simple harmonic...Ch. 3.4 - Modeling Data The table shows the temperatures T...Ch. 3.4 - Modeling Data The normal daily maximum...Ch. 3.4 - Biology The number N of bacteria in a culture...Ch. 3.4 - Depreciation The value V of a machine t years...Ch. 3.4 - Finding a Pattern Consider the function f(x)=sinx,...Ch. 3.4 - Conjecture Let f be a differentiable function of...Ch. 3.4 - Think About It Let r(x)=f(g(x)) and s(x)=g(f(x)),...Ch. 3.4 - Using Trigonometric Functions (a) Find the...Ch. 3.4 - Even and Odd Functions (a) Show that the...Ch. 3.4 - l72. Proof Let u be a differentiate function of x....Ch. 3.4 - Using Absolute Value In Exercises 173-176, use the...Ch. 3.4 - Using Absolute Value In Exercises 173-176, use the...Ch. 3.4 - Using Absolute Value In Exercises 173-176, use the...Ch. 3.4 - Using Absolute Value In Exercises 173-176, use the...Ch. 3.4 - Linear and Quadratic Approximations The linear and...Ch. 3.4 - Linear and Quadratic Approximations The linear and...Ch. 3.4 - Linear and Quadratic Approximations The linear and...Ch. 3.4 - Linear and Quadratic Approximations The linear and...Ch. 3.4 - True or False? In Exercises 181-184, determine...Ch. 3.4 - True or False? In Exercises 181-184, determine...Ch. 3.4 - True or False? In Exercises 181-184, determine...Ch. 3.4 - True or False? In Exercises 181-184, determine...Ch. 3.4 - Let f(x)=a1sinx+a2sin2x++annx where a1,a2,,an are...Ch. 3.4 - Let k be a fixed positive integer. The n th...Ch. 3.5 - Explicit and Implicit Functions Describe the...Ch. 3.5 - Implicit Differentiation Explain when you have to...Ch. 3.5 - Chain Rule How is the Chain Rule applied when...Ch. 3.5 - Logarithmic Differentiation When is it beneficial...Ch. 3.5 - Finding a Derivative In Exercises 5-26, find dy/dx...Ch. 3.5 - Finding a Derivative In Exercises 5-26, find dy/dx...Ch. 3.5 - Finding a Derivative In Exercises 5-26, find dy/dx...Ch. 3.5 - Finding a Derivative In Exercises 5-26, find dy/dx...Ch. 3.5 - Finding a Derivative In Exercises 5-26, find dy/dx...Ch. 3.5 - Finding a Derivative In Exercises 5-26, find dy/dx...Ch. 3.5 - Finding a Derivative In Exercises 5-26, find dy/dx...Ch. 3.5 - Finding a Derivative In Exercises 5-26, find dy/dx...Ch. 3.5 - Finding a Derivative In Exercises 5-26, find dy/dx...Ch. 3.5 - Finding a Derivative In Exercises 5-26, find dy/dx...Ch. 3.5 - Finding a Derivative In Exercises 5-26, find dy/dx...Ch. 3.5 - Finding a Derivative In Exercises 5-26, find dy/dx...Ch. 3.5 - 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Finding the Slope of a Graph In Exercises 31-38....Ch. 3.5 - Finding the Slope of a Graph En Exercises31-38....Ch. 3.5 - Finding the Slope of a Graph En Exercises31-38....Ch. 3.5 - Finding the Slope of a Graph In Exercises 31-38....Ch. 3.5 - Finding the Slope of a graph in Exercises31-38....Ch. 3.5 - Famous Curves In Exercises 39-J2, find the slope...Ch. 3.5 - Famous Curves In Exercises 39-42, find the slope...Ch. 3.5 - Famous Curves In Exercises 39-42, find the slope...Ch. 3.5 - Famous Curves In Exercises 39-J2, find the slope...Ch. 3.5 - Famous Curves In Exercises 45-48, find an equation...Ch. 3.5 - Famous Curves In Exercises 45-48, find an equation...Ch. 3.5 - Famous Curves In Exercises 45-48, find an equation...Ch. 3.5 - Famous Curves In Exercises 45-48, find an equation...Ch. 3.5 - Famous Curves In Exercises 45-48, find an equation...Ch. 3.5 - Famous Curves In Exercises 45-48, find an equation...Ch. 3.5 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.5 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.5 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.5 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.5 - Implicit and Explicit Forms Write two different...Ch. 3.5 - Think About all Explain why the derivative of...Ch. 3.5 - Ellipse (a) Use implicit differentiation to find...Ch. 3.5 - Hyperbola (a) Use implicit differentiation to find...Ch. 3.5 - Determining a Differentiate Function In Exercises...Ch. 3.5 - Determining a Differentiate Function In Exercises...Ch. 3.5 - Finding a Second Derivative In Exercises 59-62,...Ch. 3.5 - Finding a Second Derivative In Exercises 59-62,...Ch. 3.5 - Finding a Second Derivative In Exercises 59-62,...Ch. 3.5 - Finding a Second Derivative In Exercises 59-62,...Ch. 3.5 - Tangent Lines and Normal Lines In Exercises 63 and...Ch. 3.5 - Tangent Lines and Normal Lines In Exercises 63 and...Ch. 3.5 - Normal Lines Show that the normal line at any...Ch. 3.5 - Circle Two circles of radius 4 are tangent to the...Ch. 3.5 - Vertical and Horizontal Tangent Lines In Exercises...Ch. 3.5 - Vertical and Horizontal Tangent Lines In Exercises...Ch. 3.5 - Logarithmic Differentiation In Exercises 69-80,...Ch. 3.5 - Logarithmic Differentiation In Exercises 69-80,...Ch. 3.5 - Logarithmic Differentiation In Exercises 69-80,...Ch. 3.5 - Logarithmic Differentiation In Exercises 69-80,...Ch. 3.5 - Logarithmic Differentiation In Exercises 69-80,...Ch. 3.5 - Logarithmic Differentiation In Exercises 69-80,...Ch. 3.5 - Logarithmic Differentiation In Exercises 69-80,...Ch. 3.5 - Logarithmic Differentiation In Exercises 69-80,...Ch. 3.5 - Logarithmic Differentiation In Exercises 69-80,...Ch. 3.5 - Logarithmic Differentiation In Exercises 69-80,...Ch. 3.5 - Logarithmic Differentiation In Exercises 69-80,...Ch. 3.5 - Logarithmic Differentiation in Exercises 69-80,...Ch. 3.5 - Orthogonal Trajectories In Exercises 81-84, use a...Ch. 3.5 - Orthogonal Trajectories In Exercises 81-84, use a...Ch. 3.5 - Orthogonal Trajectories In Exercises 81-84, use a...Ch. 3.5 - Orthogonal Trajectories In Exercises 81-84, use a...Ch. 3.5 - Orthogonal Trajectories In Exercises 85 and 86,...Ch. 3.5 - Orthogonal Trajectories In Exercises 85 and 86,...Ch. 3.5 - True or False? Determine whether the statement is...Ch. 3.5 - HOW DO YOU SEE IT? Use the graph 10 answer the...Ch. 3.5 - Finding Equations of Tangent Lines Consider the...Ch. 3.5 - Tangent Lines and Intercepts Let t. be any tangent...Ch. 3.5 - Slope Find all points on the circle x2+y2=100...Ch. 3.5 - Proof (a) Prove (Theorem 3.3) that d/dx[xn]=nxn1...Ch. 3.5 - Tangent Lines Find equations of both tangent lines...Ch. 3.5 - Normals to a Parabola The graph shows the normal...Ch. 3.5 - Normal Lines (a) Find an equation of the normal...Ch. 3.6 - Evaluating the Derivative of an Inverse Function...Ch. 3.6 - Inverse Trigonometric Functions Are the...Ch. 3.6 - Evaluating the Derivative of an Inverse Function...Ch. 3.6 - Evaluating the Derivative of an Inverse Function...Ch. 3.6 - Evaluating the Derivative of an Inverse Function...Ch. 3.6 - Evaluating the Derivative of an Inverse Function...Ch. 3.6 - Evaluating the Derivative of an Inverse Function...Ch. 3.6 - Evaluating the Derivative of an Inverse Function...Ch. 3.6 - Evaluating the Derivative of an Inverse Function...Ch. 3.6 - Evaluating the Derivative of an Inverse Function...Ch. 3.6 - Evaluating the Derivative of an Inverse Function...Ch. 3.6 - Evaluating the Derivative of an Inverse Function...Ch. 3.6 - Graphs of Inverse Functions Have Reciprocal Slopes...Ch. 3.6 - Graphs of Inverse Functions Have Reciprocal Slopes...Ch. 3.6 - Graphs of Inverse Functions Have Reciprocal Slopes...Ch. 3.6 - Graphs of Inverse Functions Have Reciprocal Slopes...Ch. 3.6 - Finding a Derivative In Exercises 17-42, find the...Ch. 3.6 - Finding a Derivative In Exercises 17-42, find the...Ch. 3.6 - Finding a Derivative In Exercises 17-42, find the...Ch. 3.6 - Finding a Derivative In Exercises 17-42, find the...Ch. 3.6 - Finding a Derivative In Exercises 17-42, find the...Ch. 3.6 - Finding a Derivative In Exercises 17-42, find the...Ch. 3.6 - Finding a Derivative In Exercises 17-42, find the...Ch. 3.6 - Finding a Derivative In Exercises 17-42, find the...Ch. 3.6 - Finding a Derivative In Exercises 17-42, find the...Ch. 3.6 - Finding a Derivative In Exercises 17-42, find the...Ch. 3.6 - Finding a Derivative In Exercises 17-42, find the...Ch. 3.6 - Finding a Derivative In Exercises 17-42, find the...Ch. 3.6 - Finding a Derivative In Exercises 17-42, find the...Ch. 3.6 - Finding a Derivative In Exercises 17-42, find the...Ch. 3.6 - Finding a Derivative In Exercises 17-42, find the...Ch. 3.6 - Finding a Derivative In Exercises 17-42, find the...Ch. 3.6 - Finding a Derivative In Exercises 17-42, find the...Ch. 3.6 - Finding a Derivative In Exercises 17-42, find the...Ch. 3.6 - Finding a Derivative In Exercises 17-42, find the...Ch. 3.6 - Finding a Derivative In Exercises 17-42, find the...Ch. 3.6 - Finding a Derivative In Exercises 17-42, find the...Ch. 3.6 - Finding a Derivative In Exercises 17-42, find the...Ch. 3.6 - Finding a Derivative In Exercises 17-42, find the...Ch. 3.6 - Finding a Derivative In Exercises 17-42, find the...Ch. 3.6 - Finding a Derivative In Exercises 17-42, find the...Ch. 3.6 - Finding a Derivative In Exercises 17-42, find the...Ch. 3.6 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.6 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.6 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.6 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.6 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.6 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.6 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.6 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.6 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.6 - Finding an Equation of a Tangent Line In Exercises...Ch. 3.6 - Tangent Lines Find equations of all tangent lines...Ch. 3.6 - Tangent Lines Find equations of all tangent lines...Ch. 3.6 - Linear and Quadratic Approximations The linear and...Ch. 3.6 - Linear and Quadratic Approximations The linear and...Ch. 3.6 - Linear and Quadratic Approximations The linear and...Ch. 3.6 - Linear and Quadratic Approximations The linear and...Ch. 3.6 - Finding the Slope of a Graph In Exercises 59-62,...Ch. 3.6 - Finding the Slope of a Graph In Exercises 59-62,...Ch. 3.6 - Finding the Slope of a Graph In Exercises 59-62,...Ch. 3.6 - Finding the Slope of a Graph In Exercises 59-62,...Ch. 3.6 - Finding the Equation of a Tangent Line In...Ch. 3.6 - Finding the Equation of a Tangent Line In...Ch. 3.6 - Finding the Equation of a Tangent Line In...Ch. 3.6 - Finding the Equation of a Tangent Line In...Ch. 3.6 - Exploring concepts Inverse Secant-Function...Ch. 3.6 - Derivatives of Inverse Trigonometric Functions...Ch. 3.6 - Tangent Lines Consider the tangent line j to the...Ch. 3.6 - HOW DO YOU SEE IT? Use the information in the...Ch. 3.6 - Think About It The point (1, 3) lies on the graph...Ch. 3.6 - Angular Rate of change An...Ch. 3.6 - Angular Rate of Change Repeat Exercise 73 for an...Ch. 3.6 - Angular Rate of Change In a free-fall- experiment,...Ch. 3.6 - Angular Rate of Change A television camera at...Ch. 3.6 - Angular Rate of Change An observer is standing 300...Ch. 3.6 - Angular Speed A patrol car is parked 50 feet from...Ch. 3.6 - True or False? In Exercises 79-82, determine...Ch. 3.6 - True or False? In Exercises 79-82, determine...Ch. 3.6 - True or False? In Exercises 79-82, determine...Ch. 3.6 - True or False? In Exercises 79-82, determine...Ch. 3.6 - Proof prove each differential formula (a)...Ch. 3.6 - Proof Prove that arccosx=2arctan(x1x2), |x|1Ch. 3.6 - Proof prove that arcsinx=arctan(x1x2),x1Ch. 3.6 - Proof Show that the function...Ch. 3.7 - Related-Rate Equation What is a related rate...Ch. 3.7 - Related Rates In your own words, state the...Ch. 3.7 - Using Related Rates In Exercises 3-6, assume that...Ch. 3.7 - Using Related Rates In Exercises 3-6, assume that...Ch. 3.7 - Using Related Rates In Exercises 3-6, assume that...Ch. 3.7 - Using Related Rates In Exercises 3-6, assume that...Ch. 3.7 - Moving Point In Exercises 7-10, a point is moving...Ch. 3.7 - Moving Point In Exercises 7-10, a point is moving...Ch. 3.7 - Moving Point In Exercises 7-10, a point is moving...Ch. 3.7 - Moving Point In Exercises 7-10, a point is moving...Ch. 3.7 - Area The radius r of a circle is increasing at a...Ch. 3.7 - Area The length. s of each side of an equilateral...Ch. 3.7 - Volume The radius r of a sphere is increasing at a...Ch. 3.7 - Radius A spherical balloon is inflated with gas at...Ch. 3.7 - Volume All edges of a cube are expanding at a rate...Ch. 3.7 - Surface Area All edges of a cube are expanding at...Ch. 3.7 - Height At a sand and gravel plant, sand is falling...Ch. 3.7 - Height The volume of oil in a cylindrical...Ch. 3.7 - Depth A swimming pool is 12 meters long. 6 meters...Ch. 3.7 - Depth A trough is 12 feet long and 3 feet across...Ch. 3.7 - Moving Ladder A ladder 25 feet long is leaning...Ch. 3.7 - Construction A construction worker pulls a...Ch. 3.7 - Construction A winch at the top of a 12-meter...Ch. 3.7 - Boating A boat is pulled into a dock by means of a...Ch. 3.7 - Air Traffic Control An air traffic controller...Ch. 3.7 - Air Traffic Control An airplane is flying at an...Ch. 3.7 - Sports A baseball diamond has the shape of a...Ch. 3.7 - Sports For (be baseball diamond in Exercise 27,...Ch. 3.7 - Shadow Length A man 6 feet tall walks at a rate of...Ch. 3.7 - Shadow Length Repeat Exercise 29 for a man 6 feet...Ch. 3.7 - Machine Design The endpoints of a movable rod of...Ch. 3.7 - Machine Design Repeat Exercise 31 for a position...Ch. 3.7 - Evaporation As a spherical raindrop falls, it...Ch. 3.7 - HOW DO YOU SEE IT? 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