Finding Second Derivatives In Exercises 31-34, find the second derivative of the function. f ( x ) = ( 3 + 2 x ) e − 3 x | bartleby

Calculus: An Applied Approach (Min...

10th Edition

Ron Larson

Publisher: Cengage Learning

ISBN: 9781305860919

Chapter 4.3, Problem 34E

Textbook Problem

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Finding Second Derivatives In Exercises 31-34, find the second derivative of the function.

f ( x ) = ( 3 + 2 x ) e − 3 x

To determine

To calculate: The second derivative of the function f(x)=(3+2x)e−3x.

Explanation of Solution

Given Information:

The provided function is f(x)=(3+2x)e−3x.

Formula used:

The derivative of natural exponential

ddxeu=eududx, where, u is a differentiable function of x.

The product rule of derivative;

ddx[f(x)⋅g(x)]=ddxf(x)⋅g(x)+f(x)⋅ddxg(x).

Constant multiple rule of derivative of function;

f'(cx)=cf'(x)

Where c is constant.

Calculation:

Consider the function, f(x)=(3+2x)e−3x

Calculate the first derivative as,

f′(x)=ddx[(3+2x)e−3x]

Apply the product rule as,

f′(x)=ddx(3+2x)⋅e−3x+(3+2x)⋅ddx(e−3x)=(ddx(3)+ddx(2x))⋅e−3x+(3+2x)⋅ddx(e−3x)

Now apply the derivative of natural exponential, the constant multiple rule and the above formula as,

f′(x)=(0+2ddx(x))⋅e−3x+(3+2x)⋅e−3xddx(−3x)=(0+2⋅1)⋅e−3x+(3+2x)⋅e−3x(−3)ddx(x)=2e−3x−3(3+2x)e−3x⋅1=2e−3x−(9+6x)e−3x

Take out e−3x as common,

f′(x)=e−3x(

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Finding an Equation of a Tangent Line In Exercises...Ch. 4.5 - Finding an Equation of a Tangent Line In Exercises...Ch. 4.5 - Finding an Equation of a Tangent Line In Exercises...Ch. 4.5 - Finding an Equation of a Tangent Line In Exercises...Ch. 4.5 - Finding an Equation of a Tangent Line In Exercises...Ch. 4.5 - Finding an Equation of a Tangent Line In Exercises...Ch. 4.5 - Finding Derivatives Implicitly In Exercises 53-56,...Ch. 4.5 - Finding Derivatives Implicitly In Exercises 53-56,...Ch. 4.5 - Finding Derivatives Implicitly In Exercises 53-56,...Ch. 4.5 - Finding Derivatives Implicitly In Exercises 53-56,...Ch. 4.5 - Finding Second Derivatives In Exercises 57-62,...Ch. 4.5 - Finding Second Derivatives In Exercises 57-62,...Ch. 4.5 - Finding Second Derivatives In Exercises 57-62,...Ch. 4.5 - Finding Second Derivatives In Exercises 57-62,...Ch. 4.5 - Finding Second Derivatives In Exercises 57-62,...Ch. 4.5 - Finding Second Derivatives In Exercises 57-62,...Ch. 4.5 - Sound Intensity The relationship between the...Ch. 4.5 - Chemistry The temperatures T (inF) at which water...Ch. 4.5 - Finding Relative Extrema In Exercises 65-68, find...Ch. 4.5 - Finding Relative Extrema In Exercises 65-68, find...Ch. 4.5 - Finding Relative Extrema In Exercises 65-68, find...Ch. 4.5 - Finding Relative Extrema In Exercises 65-68, find...Ch. 4.5 - Analyzing a Graph In Exercises 6972, analyze and...Ch. 4.5 - Analyzing a Graph In Exercises 6972, analyze and...Ch. 4.5 - Analyzing a Graph In Exercises 69-72, analyze and...Ch. 4.5 - Analyzing a Graph In Exercises 69-72, analyze and...Ch. 4.5 - Demand In Exercises 73 and 74, find dx/dp for the...Ch. 4.5 - Demand In Exercises 73 and 74, find dx/dp for the...Ch. 4.5 - Demand Solve the demand function in Exercise 73...Ch. 4.5 - Demand Solve the demand function in Exercise 74...Ch. 4.5 - Minimum Average Cost The cost of producing x units...Ch. 4.5 - Minimum Average Cost The cost of producing x units...Ch. 4.5 - Consumer Trends The numbers of employees E (in...Ch. 4.5 - HOW DO YOU SEE IT? The graph shows the temperature...Ch. 4.5 - Home Mortgage The term t (in years) of a $200,000...Ch. 4.5 - Earthquake Intensity On the Richter scale, the...Ch. 4.5 - Learning Theory Students in a learning theory...Ch. 4.6 - Checkpoint 1 Worked-out solution available at...Ch. 4.6 - Checkpoint 2 Worked-out solution available at...Ch. 4.6 - Checkpoint 3 Worked-out solution available at...Ch. 4.6 - Checkpoint 4 Worked-out solution available at...Ch. 4.6 - In Exercises 1-4, solve the equation for k....Ch. 4.6 - In Exercises 1-4, solve the equation for k....Ch. 4.6 - In Exercises 1-4, solve the equation for k....Ch. 4.6 - In Exercises 1-4, solve the equation for k....Ch. 4.6 - In Exercises 5-8, find the derivative of the...Ch. 4.6 - In Exercises 5-8, find the derivative of the...Ch. 4.6 - In Exercises 5-8, find the derivative of the...Ch. 4.6 - In Exercises 5-8, find the derivative of the...Ch. 4.6 - In Exercises 9-12, simplify the expression. eln4 Ch. 4.6 - In Exercises 9-12, simplify the expression....Ch. 4.6 - In Exercises 9-12, simplify the expression....Ch. 4.6 - In Exercises 9-12, simplify the expression....Ch. 4.6 - Modeling Exponential Growth and Decay In Exercises...Ch. 4.6 - Modeling Exponential Growth and Decay In Exercises...Ch. 4.6 - Modeling Exponential Growth and Decay In Exercises...Ch. 4.6 - Modeling Exponential Growth and Decay In Exercises...Ch. 4.6 - Modeling Exponential Growth and Decay In Exercises...Ch. 4.6 - Modeling Exponential Growth and Decay In Exercises...Ch. 4.6 - Determining Exponential Growth and Decay In...Ch. 4.6 - Determining Exponential Growth and Decay In...Ch. 4.6 - Determining Exponential Growth and Decay In...Ch. 4.6 - Determining Exponential Growth and Decay In...Ch. 4.6 - Modeling Radioactive Decay In Exercises 11-16,...Ch. 4.6 - Modeling Radioactive Decay In Exercises 11-16,...Ch. 4.6 - Modeling Radioactive Decay In Exercises 11-16,...Ch. 4.6 - Modeling Radioactive Decay In Exercises 11-16,...Ch. 4.6 - Modeling Radioactive Decay In Exercises 11-16,...Ch. 4.6 - Modeling Radioactive Decay In Exercises 11-16,...Ch. 4.6 - Radioactive Decay What percent of a present amount...Ch. 4.6 - Radioactive Decay Find the half-life of a...Ch. 4.6 - Carbon Dating Carbon-14 (14C) dating assumes that...Ch. 4.6 - Carbon Dating Repeat Exercise 19 for a piece of...Ch. 4.6 - Finding Exponential Models In Exercises 21 and 22,...Ch. 4.6 - Finding Exponential Models In Exercises 21 and 22,...Ch. 4.6 - Population Growth The number of a certain type of...Ch. 4.6 - HOW DO YOU SEE IT? The graph shows the populations...Ch. 4.6 - Modeling Compound Interest In Exercises 25-32,...Ch. 4.6 - Modeling Compound Interest In Exercises 25-32,...Ch. 4.6 - Modeling Compound Interest In Exercises 25-32,...Ch. 4.6 - Modeling Compound Interest In Exercises 25-32,...Ch. 4.6 - Modeling Compound Interest In Exercises 25-32,...Ch. 4.6 - Modeling Compound Interest In Exercises 25-32,...Ch. 4.6 - Modeling Compound Interest In Exercises 25-32,...Ch. 4.6 - Modeling Compound Interest In Exercises 25-32,...Ch. 4.6 - Finding Present Value In Exercises 33 and 34,...Ch. 4.6 - Finding Present Value In Exercises 33 and 34,...Ch. 4.6 - Effective Rate The effective rate of interest reff...Ch. 4.6 - Effective Rate Use the results of Exercise 35 to...Ch. 4.6 - Investment: Rule of 70 Verify that the time...Ch. 4.6 - Investment: Rule of 70 Use the Rule of 70 from...Ch. 4.6 - Depreciation A sports utility vehicle that costs...Ch. 4.6 - Population The table shows the populations P (in...Ch. 4.6 - Sales The cumulative sales S (in thousands of...Ch. 4.6 - Sales The cumulative sales S (in thousands of...Ch. 4.6 - Learning Curve The management of a factory finds...Ch. 4.6 - Learning Curve The management in Exercise 43...Ch. 4.6 - Revenue A small business assumes that the demand...Ch. 4.6 - Revenue Repeat Exercise 45 given that when...Ch. 4 - Applying Properties of Exponents In Exercises 1...Ch. 4 - Applying Properties of Exponents In Exercises 1...Ch. 4 - Graphing Exponential Functions In Exercises 3-8,...Ch. 4 - Graphing Exponential Functions In Exercises 3-8,...Ch. 4 - Graphing Exponential Functions In Exercises 3-8,...Ch. 4 - Graphing Exponential Functions In Exercises 3-8,...Ch. 4 - Graphing Exponential Functions In Exercises 3-8,...Ch. 4 - Graphing Exponential Functions In Exercises 3-8,...Ch. 4 - Population Growth The resident populations P (in...Ch. 4 - Revenue The revenues R (in millions of dollars)...Ch. 4 - Property Value A piece of property sells for...Ch. 4 - Inflation Rate With an annual rate of inflation of...Ch. 4 - Applying Properties of Exponents In Exercises 13...Ch. 4 - Applying Properties of Exponents In Exercises 13...Ch. 4 - Graphing Natural Exponential Functions In...Ch. 4 - Graphing Natural Exponential Functions In...Ch. 4 - Graphing Natural Exponential Functions In...Ch. 4 - Graphing Natural Exponential Functions In...Ch. 4 - Finding Account Balances In Exercises 19-22,...Ch. 4 - Finding Account Balances In Exercises 19-22,...Ch. 4 - Finding Account Balances In Exercises 19-22,...Ch. 4 - Finding Account Balances In Exercises 19-22,...Ch. 4 - Comparing Account Balances In Exercises 23 and 24,...Ch. 4 - Comparing Account Balances In Exercises 23 and 24,...Ch. 4 - Effective Rate Find the effective rate of interest...Ch. 4 - Effective Rate Find the effective rate of interest...Ch. 4 - Present Value How much should be deposited in an...Ch. 4 - Present Value How much should be deposited in an...Ch. 4 - Demand The demand function for a product is...Ch. 4 - Demand The demand function for a product is...Ch. 4 - Profit The net profits P (in millions of dollars)...Ch. 4 - Population The populations P (in thousands) of...Ch. 4 - Biology A lake is stocked with 500 fish, and the...Ch. 4 - Medicine On a college campus of 5000 students, the...Ch. 4 - Differentiating Exponential Functions In Exercises...Ch. 4 - Differentiating Exponential Functions In Exercises...Ch. 4 - Differentiating Exponential Functions In Exercises...Ch. 4 - Differentiating Exponential Functions In Exercises...Ch. 4 - Differentiating Exponential Functions In Exercises...Ch. 4 - Differentiating Exponential Functions In Exercises...Ch. 4 - Finding an Equation of a Tangent Line In Exercises...Ch. 4 - Finding an Equation of a Tangent Line In Exercises...Ch. 4 - Finding an Equation of a Tangent Line In Exercises...Ch. 4 - Finding an Equation of a Tangent Line In Exercises...Ch. 4 - Analyzing a Graph In Exercises 4548, analyze and...Ch. 4 - Analyzing a Graph In Exercises 4548, analyze and...Ch. 4 - Analyzing a Graph In Exercises 4548, analyze and...Ch. 4 - Analyzing a Graph In Exercises 4548, analyze and...Ch. 4 - Logarithmic and Exponential Forms of Equations In...Ch. 4 - Logarithmic and Exponential Forms of Equations In...Ch. 4 - Logarithmic and Exponential Forms of Equations In...Ch. 4 - Logarithmic and Exponential Forms of Equations In...Ch. 4 - Graphing Logarithmic Functions In Exercises 5356,...Ch. 4 - Graphing Logarithmic Functions In Exercises 5356,...Ch. 4 - Graphing Logarithmic Functions In Exercises 5356,...Ch. 4 - Graphing Logarithmic Functions In Exercises 5356,...Ch. 4 - Expanding Logarithmic Expressions In Exercises...Ch. 4 - Expanding Logarithmic Expressions In Exercises...Ch. 4 - Expanding Logarithmic Expressions In Exercises...Ch. 4 - Expanding Logarithmic Expressions In Exercises...Ch. 4 - Expanding Logarithmic Expressions In Exercises...Ch. 4 - Expanding Logarithmic Expressions In Exercises...Ch. 4 - Condensing Logarithmic Expressions In Exercises...Ch. 4 - Condensing Logarithmic Expressions In Exercises...Ch. 4 - Condensing Logarithmic Expressions In Exercises...Ch. 4 - Condensing Logarithmic Expressions In Exercises...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Compound Interest A deposit of $400 is made in an...Ch. 4 - Hourly Earnings The average hourly wages w (in...Ch. 4 - Learning Theory Students in a psychology...Ch. 4 - Demand The demand function for a product is given...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Evaluating Logarithms In Exercises 99-102,...Ch. 4 - Evaluating Logarithms In Exercises 99-102,...Ch. 4 - Evaluating Logarithms In Exercises 99-102,...Ch. 4 - Evaluating Logarithms In Exercises 99-102,...Ch. 4 - Changing Bases to Evaluate Logarithms In Exercises...Ch. 4 - Changing Bases to Evaluate Logarithms In Exercises...Ch. 4 - Changing Bases to Evaluate Logarithms In Exercises...Ch. 4 - Changing Bases to Evaluate Logarithms In Exercises...Ch. 4 - Differentiating Functions of Other Bases In...Ch. 4 - Differentiating Functions of Other Bases In...Ch. 4 - Differentiating Functions of Other Bases In...Ch. 4 - Differentiating Functions of Other Bases In...Ch. 4 - Differentiating Functions of Other Bases In...Ch. 4 - Differentiating Functions of Other Bases In...Ch. 4 - Finding Relative Extrema In Exercises 113-116,...Ch. 4 - Finding Relative Extrema In Exercises 113-116,...Ch. 4 - Finding Relative Extrema In Exercises 113-116,...Ch. 4 - Finding Relative Extrema In Exercises 113-116,...Ch. 4 - Music The numbers of music albums downloaded D (in...Ch. 4 - Minimum Average Cost The cost of producing x units...Ch. 4 - Modeling Exponential Growth and Decay In Exercises...Ch. 4 - Modeling Exponential Growth and Decay In Exercises...Ch. 4 - Modeling Radioactive Decay In Exercises 121-126,...Ch. 4 - Modeling Radioactive Decay In Exercises 121-126,...Ch. 4 - Modeling Radioactive Decay In Exercises 121-126,...Ch. 4 - Modeling Radioactive Decay In Exercises 121-126,...Ch. 4 - Modeling Radioactive Decay In Exercises 121-126,...Ch. 4 - Modeling Radioactive Decay In Exercises 121-126,...Ch. 4 - Modeling Compound Interest In Exercises 127-130,...Ch. 4 - Modeling Compound Interest In Exercises 127-130,...Ch. 4 - Modeling Compound Interest In Exercises 127-130,...Ch. 4 - Modeling Compound Interest In Exercises 127-130,...Ch. 4 - Medical Science Soon after an injection, the...Ch. 4 - Population Growth The number of a certain type of...Ch. 4 - In Exercises 14, use the properties of exponents...Ch. 4 - In Exercises 14, use the properties of exponents...Ch. 4 - In Exercises 14, use the properties of exponents...Ch. 4 - In Exercises 14, use the properties of exponents...Ch. 4 - In Exercises 5-10, sketch the graph of the...Ch. 4 - In Exercises 5-10, sketch the graph of the...Ch. 4 - In Exercises 5-10, sketch the graph of the...Ch. 4 - In Exercises 5-10, sketch the graph of the...Ch. 4 - In Exercises 5-10, sketch the graph of the...Ch. 4 - In Exercises 5-10, sketch the graph of the...Ch. 4 - In Exercises 1113, use the properties of...Ch. 4 - In Exercises 1113, use the properties of...Ch. 4 - In Exercises 1113, use the properties of...Ch. 4 - In Exercises 1416, use the properties of...Ch. 4 - In Exercises 1416, use the properties of...Ch. 4 - In Exercises 1416, use the properties of...Ch. 4 - In Exercises 17-19, solve for x. ln3x=3.5 Ch. 4 - In Exercises 17-19, solve for x. 10e2x+1=900 Ch. 4 - In Exercises 17-19, solve for x. 50(1.06)x=1500 Ch. 4 - A deposit of $500 is made in an account that earns...Ch. 4 - In Exercises 21-26, find the derivative of the...Ch. 4 - In Exercises 21-26, find the derivative of the...Ch. 4 - In Exercises 21-26, find the derivative of the...Ch. 4 - In Exercises 21-26, find the derivative of the...Ch. 4 - In Exercises 21-26, find the derivative of the...Ch. 4 - In Exercises 21-26, find the derivative of the...Ch. 4 - The sales S (in billions of dollars) of organic...Ch. 4 - What percent of a present amount of radioactive...Ch. 4 - A population is growing continuously at the rate...

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