Explanation Given: Function f ( x ) = x 2 ( x 3 + 5 ) . Formula Used: Product rule of derivative is as follows: ( f . g ) ′ = f ′ . g + f . g ′ Calculation: Function is given as f ( x ) = x 2 ( x 3 + 5 ) . We need to calculate the derivative of above function. Method 1: Calculating derivative using product rule f ′ ( x ) = [ d d x ( x 2 ) × ( x 3 + 5 ) ] + [ ( x 2 ) × d d x ( x 3 + 5 ) ] f ′ ( x ) = [ ( 2 x ) ( x 3 + 5 ) ] + [ ( x 2 ) ( 3 x 2 ) ] f ′ ( x ) = 2 x 4 + 10 x + 3 x 4 f ′ ( x ) = 5 x 4 + 10 x Method 2: Multiplying before calculating the derivative

6th Edition

ISBN: 9781119399322

Author: Hughes-Hallett

Publisher: WILEY

See similar textbooks

Videos

Question

Chapter 3.4, Problem 1P

To determine

To find f′(x) in two ways − by using the product rule and by multiplying out before taking the derivative.

f(x)=x2(x3+5)

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 3.3, Problem 53P

Chapter 3.4, Problem 2P

Students have asked these similar questions

3. We'd like to know the first time when the population reaches 7000 people. First, graph the function from part (a) on your calculator or Desmos. In the same window, graph the line y = 7000. Notice that you will need to adjust your window so that you can see values as big as 7000! Investigate the intersection of the two graphs. (This video shows you how to find the intersection on your calculator, or in Desmos just hover the cursor over the point.) At what value t> 0 does the line intersect with your exponential function? Round your answer to two decimal places. (You don't need to show work for this part.) (2 points)

Suppose the planet of Tattooine currently has a population of 6500 people and an annual growth rate of 0.35%. Use this information for all the problems below. 1. Find an exponential function f(t) that gives the population of Tattooine t years from now. (3 points)

A house was valued at $95,000 in the year 1988. The value appreciated to $170,000 by the year 2007. A) If the value is growing exponentially, what was the annual growth rate between 1988 and 2007? Round the growth rate to 4 decimal places. r = B) What is the correct answer to part A written in percentage form? r = 3 %.

Chapter 3 Solutions

APPLIED CALCULUS (WILEY PLUS)

Ch. 3.1 - Prob. 1P Ch. 3.1 - Prob. 2P Ch. 3.1 - Prob. 3P Ch. 3.1 - Prob. 4P Ch. 3.1 - Prob. 5P Ch. 3.1 - Prob. 6P Ch. 3.1 - Prob. 7P Ch. 3.1 - Prob. 8P Ch. 3.1 - Prob. 9P Ch. 3.1 - Prob. 10P

Ch. 3.1 - Prob. 11P Ch. 3.1 - Prob. 12P Ch. 3.1 - Prob. 13P Ch. 3.1 - Prob. 14P Ch. 3.1 - Prob. 15P Ch. 3.1 - Prob. 16P Ch. 3.1 - Prob. 17P Ch. 3.1 - Prob. 18P Ch. 3.1 - Prob. 19P Ch. 3.1 - Prob. 20P Ch. 3.1 - Prob. 21P Ch. 3.1 - Prob. 22P Ch. 3.1 - Prob. 23P Ch. 3.1 - Prob. 24P Ch. 3.1 - Prob. 25P Ch. 3.1 - Prob. 26P Ch. 3.1 - Prob. 27P Ch. 3.1 - Prob. 28P Ch. 3.1 - Prob. 29P Ch. 3.1 - Prob. 30P Ch. 3.1 - Prob. 31P Ch. 3.1 - Prob. 32P Ch. 3.1 - Prob. 33P Ch. 3.1 - Prob. 34P Ch. 3.1 - Prob. 35P Ch. 3.1 - Prob. 36P Ch. 3.1 - Prob. 37P Ch. 3.1 - Prob. 38P Ch. 3.1 - Prob. 39P Ch. 3.1 - Prob. 40P Ch. 3.1 - Prob. 41P Ch. 3.1 - Prob. 42P Ch. 3.1 - Prob. 43P Ch. 3.1 - Prob. 44P Ch. 3.1 - Prob. 45P Ch. 3.1 - Prob. 46P Ch. 3.1 - Prob. 47P Ch. 3.1 - Prob. 48P Ch. 3.1 - Prob. 49P Ch. 3.1 - Prob. 50P Ch. 3.1 - Prob. 51P Ch. 3.1 - Prob. 52P Ch. 3.1 - Prob. 53P Ch. 3.1 - Prob. 54P Ch. 3.1 - Prob. 55P Ch. 3.1 - Prob. 56P Ch. 3.1 - Prob. 57P Ch. 3.1 - Prob. 58P Ch. 3.1 - Prob. 59P Ch. 3.1 - Prob. 60P Ch. 3.1 - Prob. 61P Ch. 3.1 - Prob. 62P Ch. 3.1 - Prob. 63P Ch. 3.1 - Prob. 64P Ch. 3.1 - Prob. 65P Ch. 3.1 - Prob. 66P Ch. 3.1 - Prob. 67P Ch. 3.1 - Prob. 68P Ch. 3.1 - Prob. 69P Ch. 3.1 - Prob. 70P Ch. 3.1 - Prob. 71P Ch. 3.1 - Prob. 72P Ch. 3.1 - Prob. 73P Ch. 3.1 - Prob. 74P Ch. 3.1 - Prob. 75P Ch. 3.1 - Prob. 76P Ch. 3.1 - Prob. 77P Ch. 3.1 - Prob. 78P Ch. 3.1 - Prob. 79P Ch. 3.2 - Prob. 1P Ch. 3.2 - Prob. 2P Ch. 3.2 - Prob. 3P Ch. 3.2 - Prob. 4P Ch. 3.2 - Prob. 5P Ch. 3.2 - Prob. 6P Ch. 3.2 - Prob. 7P Ch. 3.2 - Prob. 8P Ch. 3.2 - Prob. 9P Ch. 3.2 - Prob. 10P Ch. 3.2 - Prob. 11P Ch. 3.2 - Prob. 12P Ch. 3.2 - Prob. 13P Ch. 3.2 - Prob. 14P Ch. 3.2 - Prob. 15P Ch. 3.2 - Prob. 16P Ch. 3.2 - Prob. 17P Ch. 3.2 - Prob. 18P Ch. 3.2 - Prob. 19P Ch. 3.2 - Prob. 20P Ch. 3.2 - Prob. 21P Ch. 3.2 - Prob. 22P Ch. 3.2 - Prob. 23P Ch. 3.2 - Prob. 24P Ch. 3.2 - Prob. 25P Ch. 3.2 - Prob. 26P Ch. 3.2 - Prob. 27P Ch. 3.2 - Prob. 28P Ch. 3.2 - Prob. 29P Ch. 3.2 - Prob. 30P Ch. 3.2 - Prob. 31P Ch. 3.2 - Prob. 32P Ch. 3.2 - Prob. 33P Ch. 3.2 - Prob. 34P Ch. 3.2 - Prob. 35P Ch. 3.2 - Prob. 36P Ch. 3.2 - Prob. 37P Ch. 3.2 - Prob. 38P Ch. 3.2 - Prob. 39P Ch. 3.2 - Prob. 40P Ch. 3.2 - Prob. 41P Ch. 3.2 - Prob. 42P Ch. 3.2 - Prob. 43P Ch. 3.2 - Prob. 44P Ch. 3.2 - Prob. 45P Ch. 3.2 - Prob. 46P Ch. 3.2 - Prob. 47P Ch. 3.2 - Prob. 48P Ch. 3.2 - Prob. 49P Ch. 3.2 - Prob. 50P Ch. 3.2 - Prob. 51P Ch. 3.2 - Prob. 52P Ch. 3.2 - Prob. 53P Ch. 3.2 - Prob. 54P Ch. 3.2 - Prob. 55P Ch. 3.2 - Prob. 56P Ch. 3.2 - Prob. 57P Ch. 3.2 - Prob. 58P Ch. 3.2 - Prob. 59P Ch. 3.2 - Prob. 60P Ch. 3.3 - Prob. 1P Ch. 3.3 - Prob. 2P Ch. 3.3 - Prob. 3P Ch. 3.3 - Prob. 4P Ch. 3.3 - Prob. 5P Ch. 3.3 - Prob. 6P Ch. 3.3 - Prob. 7P Ch. 3.3 - Prob. 8P Ch. 3.3 - Prob. 9P Ch. 3.3 - Prob. 10P Ch. 3.3 - Prob. 11P Ch. 3.3 - Prob. 12P Ch. 3.3 - Prob. 13P Ch. 3.3 - Prob. 14P Ch. 3.3 - Prob. 15P Ch. 3.3 - Prob. 16P Ch. 3.3 - Prob. 17P Ch. 3.3 - Prob. 18P Ch. 3.3 - Prob. 19P Ch. 3.3 - Prob. 20P Ch. 3.3 - Prob. 21P Ch. 3.3 - Prob. 22P Ch. 3.3 - Prob. 23P Ch. 3.3 - Prob. 24P Ch. 3.3 - Prob. 25P Ch. 3.3 - Prob. 26P Ch. 3.3 - Prob. 27P Ch. 3.3 - Prob. 28P Ch. 3.3 - Prob. 29P Ch. 3.3 - Prob. 30P Ch. 3.3 - Prob. 31P Ch. 3.3 - Prob. 32P Ch. 3.3 - Prob. 33P Ch. 3.3 - Prob. 34P Ch. 3.3 - Prob. 35P Ch. 3.3 - Prob. 36P Ch. 3.3 - Prob. 37P Ch. 3.3 - Prob. 38P Ch. 3.3 - Prob. 39P Ch. 3.3 - Prob. 40P Ch. 3.3 - Prob. 41P Ch. 3.3 - Prob. 42P Ch. 3.3 - Prob. 43P Ch. 3.3 - Prob. 44P Ch. 3.3 - Prob. 45P Ch. 3.3 - Prob. 46P Ch. 3.3 - Prob. 47P Ch. 3.3 - Prob. 48P Ch. 3.3 - Prob. 49P Ch. 3.3 - Prob. 50P Ch. 3.3 - Prob. 51P Ch. 3.3 - Prob. 52P Ch. 3.3 - Prob. 53P Ch. 3.4 - Prob. 1P Ch. 3.4 - Prob. 2P Ch. 3.4 - Prob. 3P Ch. 3.4 - Prob. 4P Ch. 3.4 - Prob. 5P Ch. 3.4 - Prob. 6P Ch. 3.4 - Prob. 7P Ch. 3.4 - Prob. 8P Ch. 3.4 - Prob. 9P Ch. 3.4 - Prob. 10P Ch. 3.4 - Prob. 11P Ch. 3.4 - Prob. 12P Ch. 3.4 - Prob. 13P Ch. 3.4 - Prob. 14P Ch. 3.4 - Prob. 15P Ch. 3.4 - Prob. 16P Ch. 3.4 - Prob. 17P Ch. 3.4 - Prob. 18P Ch. 3.4 - Prob. 19P Ch. 3.4 - Prob. 20P Ch. 3.4 - Prob. 21P Ch. 3.4 - Prob. 22P Ch. 3.4 - Prob. 23P Ch. 3.4 - Prob. 24P Ch. 3.4 - Prob. 25P Ch. 3.4 - Prob. 26P Ch. 3.4 - Prob. 27P Ch. 3.4 - Prob. 28P Ch. 3.4 - Prob. 29P Ch. 3.4 - Prob. 30P Ch. 3.4 - Prob. 31P Ch. 3.4 - Prob. 32P Ch. 3.4 - Prob. 33P Ch. 3.4 - Prob. 34P Ch. 3.4 - Prob. 35P Ch. 3.4 - Prob. 36P Ch. 3.4 - Prob. 37P Ch. 3.4 - Prob. 38P Ch. 3.4 - Prob. 39P Ch. 3.4 - Prob. 40P Ch. 3.4 - Prob. 41P Ch. 3.4 - Prob. 42P Ch. 3.4 - Prob. 43P Ch. 3.4 - Prob. 44P Ch. 3.4 - Prob. 45P Ch. 3.4 - Prob. 46P Ch. 3.4 - Prob. 47P Ch. 3.4 - Prob. 48P Ch. 3.4 - Prob. 49P Ch. 3.4 - Prob. 50P Ch. 3.4 - Prob. 51P Ch. 3.4 - Prob. 52P Ch. 3.5 - Prob. 1P Ch. 3.5 - Prob. 2P Ch. 3.5 - Prob. 3P Ch. 3.5 - Prob. 4P Ch. 3.5 - Prob. 5P Ch. 3.5 - Prob. 6P Ch. 3.5 - Prob. 7P Ch. 3.5 - Prob. 8P Ch. 3.5 - Prob. 9P Ch. 3.5 - Prob. 10P Ch. 3.5 - Prob. 11P Ch. 3.5 - Prob. 12P Ch. 3.5 - Prob. 13P Ch. 3.5 - Prob. 14P Ch. 3.5 - Prob. 15P Ch. 3.5 - Prob. 16P Ch. 3.5 - Prob. 17P Ch. 3.5 - Prob. 18P Ch. 3.5 - Prob. 19P Ch. 3.5 - Prob. 20P Ch. 3.5 - Prob. 21P Ch. 3.5 - Prob. 22P Ch. 3.5 - Prob. 23P Ch. 3.5 - Prob. 24P Ch. 3.5 - Prob. 25P Ch. 3.5 - Prob. 26P Ch. 3.5 - Prob. 27P Ch. 3.5 - Prob. 28P Ch. 3.5 - Prob. 29P Ch. 3.5 - Prob. 30P Ch. 3.5 - Prob. 31P Ch. 3.5 - Prob. 32P Ch. 3.5 - Prob. 33P Ch. 3.5 - Prob. 34P Ch. 3.5 - Prob. 35P Ch. 3.5 - Prob. 36P Ch. 3.5 - Prob. 37P Ch. 3 - Prob. 1SYU Ch. 3 - Prob. 2SYU Ch. 3 - Prob. 3SYU Ch. 3 - Prob. 4SYU Ch. 3 - Prob. 5SYU Ch. 3 - Prob. 6SYU Ch. 3 - Prob. 7SYU Ch. 3 - Prob. 8SYU Ch. 3 - Prob. 9SYU Ch. 3 - Prob. 10SYU Ch. 3 - Prob. 11SYU Ch. 3 - Prob. 12SYU Ch. 3 - Prob. 13SYU Ch. 3 - Prob. 14SYU Ch. 3 - Prob. 15SYU Ch. 3 - Prob. 16SYU Ch. 3 - Prob. 17SYU Ch. 3 - Prob. 18SYU Ch. 3 - Prob. 19SYU Ch. 3 - Prob. 20SYU Ch. 3 - Prob. 21SYU Ch. 3 - Prob. 22SYU Ch. 3 - Prob. 23SYU Ch. 3 - Prob. 24SYU Ch. 3 - Prob. 25SYU Ch. 3 - Prob. 26SYU Ch. 3 - Prob. 27SYU Ch. 3 - Prob. 28SYU Ch. 3 - Prob. 29SYU Ch. 3 - Prob. 30SYU Ch. 3 - Prob. 31SYU Ch. 3 - Prob. 32SYU Ch. 3 - Prob. 33SYU Ch. 3 - Prob. 34SYU Ch. 3 - Prob. 35SYU Ch. 3 - Prob. 36SYU Ch. 3 - Prob. 37SYU Ch. 3 - Prob. 38SYU Ch. 3 - Prob. 39SYU Ch. 3 - Prob. 40SYU Ch. 3 - Prob. 41SYU Ch. 3 - Prob. 42SYU Ch. 3 - Prob. 43SYU Ch. 3 - Prob. 44SYU Ch. 3 - Prob. 45SYU Ch. 3 - Prob. 46SYU Ch. 3 - Prob. 47SYU Ch. 3 - Prob. 48SYU Ch. 3 - Prob. 49SYU Ch. 3 - Prob. 50SYU Ch. 3 - Prob. 1FOT Ch. 3 - Prob. 2FOT Ch. 3 - Prob. 3FOT Ch. 3 - Prob. 4FOT Ch. 3 - Prob. 5FOT Ch. 3 - Prob. 6FOT Ch. 3 - Prob. 7FOT Ch. 3 - Prob. 8FOT Ch. 3 - Prob. 9FOT Ch. 3 - Prob. 10FOT Ch. 3 - Prob. 11FOT Ch. 3 - Prob. 12FOT Ch. 3 - Prob. 1FOP Ch. 3 - Prob. 2FOP Ch. 3 - Prob. 3FOP Ch. 3 - Prob. 4FOP Ch. 3 - Prob. 5FOP Ch. 3 - Prob. 6FOP Ch. 3 - Prob. 7FOP Ch. 3 - Prob. 8FOP Ch. 3 - Prob. 9FOP Ch. 3 - Prob. 10FOP Ch. 3 - Prob. 11FOP Ch. 3 - Prob. 12FOP Ch. 3 - Prob. 13FOP Ch. 3 - Prob. 14FOP Ch. 3 - Prob. 15FOP Ch. 3 - Prob. 16FOP Ch. 3 - Prob. 17FOP Ch. 3 - Prob. 18FOP Ch. 3 - Prob. 19FOP Ch. 3 - Prob. 20FOP Ch. 3 - Prob. 21FOP Ch. 3 - Prob. 22FOP Ch. 3 - Prob. 23FOP Ch. 3 - Prob. 24FOP Ch. 3 - Prob. 25FOP Ch. 3 - Prob. 26FOP Ch. 3 - Prob. 27FOP Ch. 3 - Prob. 28FOP Ch. 3 - Prob. 29FOP Ch. 3 - Prob. 30FOP Ch. 3 - Prob. 31FOP Ch. 3 - Prob. 32FOP Ch. 3 - Prob. 33FOP Ch. 3 - Prob. 34FOP Ch. 3 - Prob. 35FOP Ch. 3 - Prob. 36FOP Ch. 3 - Prob. 37FOP Ch. 3 - Prob. 38FOP Ch. 3 - Prob. 39FOP Ch. 3 - Prob. 40FOP Ch. 3 - Prob. 41FOP Ch. 3 - Prob. 42FOP Ch. 3 - Prob. 43FOP Ch. 3 - Prob. 44FOP Ch. 3 - Prob. 45FOP Ch. 3 - Prob. 46FOP Ch. 3 - Prob. 47FOP Ch. 3 - Prob. 48FOP Ch. 3 - Prob. 49FOP Ch. 3 - Prob. 50FOP Ch. 3 - Prob. 51FOP Ch. 3 - Prob. 52FOP Ch. 3 - Prob. 53FOP Ch. 3 - Prob. 54FOP Ch. 3 - Prob. 55FOP Ch. 3 - Prob. 56FOP Ch. 3 - Prob. 57FOP Ch. 3 - Prob. 58FOP Ch. 3 - Prob. 59FOP Ch. 3 - Prob. 60FOP Ch. 3 - Prob. 61FOP Ch. 3 - Prob. 62FOP Ch. 3 - Prob. 63FOP

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

To find f ′ ( x ) in two ways − by using the product rule and by multiplying out before taking the derivative. f ( x ) = x 2 ( x 3 + 5 )

Solution Summary: The author explains how to calculate the derivative of fprime(x) using the product rule and by multiplying before calculating it.

Videos

To find f′(x) in two ways − by using the product rule and by multiplying out before taking the derivative.

f(x)=x2(x3+5)

Want to see the full answer?

Chapter 3 Solutions

To find f ′ ( x ) in two ways − by using the product rule and by multiplying out before taking the derivative. f ( x ) = x 2 ( x 3 + 5 )

Solution Summary: The author explains how to calculate the derivative of fprime(x) using the product rule and by multiplying before calculating it.

Videos

To find f′(x) in two ways − by using the product rule and by multiplying out before taking the derivative. f(x)=x2(x3+5)

Want to see the full answer?

Chapter 3 Solutions

To find f′(x) in two ways − by using the product rule and by multiplying out before taking the derivative.

f(x)=x2(x3+5)