/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 115 Differentiate each function. $... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 115

Differentiate each function. $$g(x)=\left(x^{3}-8\right) \cdot \frac{x^{2}+1}{x^{2}-1}$$

Short Answer

Expert verified
Simplify the function, apply the product and quotient rules, combine results, and simplify.

Step by step solution

01

Understand the Problem

Given the function \[g(x)=\frac{(x^{3}-8)(x^{2}+1)}{x^{2}-1}\], we need to find its derivative. We will apply the product rule and quotient rule where necessary.
02

Simplify the Function

Rewrite the function for easier differentiation. Separate it into two parts: \[(x^{3}-8)\] and \[\frac{x^{2}+1}{x^{2}-1}\].
03

Apply the Product Rule

The product rule states \[\frac{d}{dx}[u \times v] = u'v + uv'\], where \[u = x^{3} - 8\] and \[v = \frac{x^{2}+1}{x^{2}-1}\]. First, find the derivatives of both parts separately.
04

Differentiate the First Part

Differentiate \[u = x^{3} - 8\] \to obtain \[u' = 3x^{2}\].
05

Differentiate the Second Part Using the Quotient Rule

The quotient rule states \[\frac{d}{dx}\bigg(\frac{f}{g}\bigg) = \frac{f'g - fg'}{g^2}\], where \[f = x^{2}+1\] and \[g = x^{2}-1\]. Calculate \[f' = 2x\] and \[g' = 2x\], then apply the quotient rule to find \[v'\].
06

Simplify the Quotient Rule Expression

Solve \[v' = \frac{(2x)(x^{2} - 1) - (x^{2}+1)(2x)}{(x^{2} - 1)^2}\]. Simplify this expression.
07

Combine the Results

Combine the differentiated parts using the product rule: \[g'(x) = (3x^{2}) \times \frac{x^{2}+1}{x^{2}-1} + (x^{3} - 8) \times v'\]. Substitute \[v'\] with the simplified expression from the previous step.
08

Final Differentiated Function

Simplify the entire expression to get the final differentiated function.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Product Rule
To differentiate functions that are multiplied together, we apply the product rule. The product rule can be stated as: \[ \frac{d}{dx} [u \times v] = u'v + uv' \]
This means if you have two functions, \( u \) and \( v \), their derivative is found by taking the derivative of the first function (\
Quotient Rule
When differentiating the quotient of two functions, we use the quotient rule:
\[ \frac{d}{dx}\bigg(\frac{f(x)}{g(x)}\bigg) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \]
This means to find the derivative of a quotient, \(f(x) / g(x)\), you need to:
  • Find the derivative of the numerator, \(f'(x)\).
  • Find the derivative of the denominator, \(g'(x)\).
  • Apply the formula by substituting \(f(x)\), \(g(x)\), \(f'(x)\), and \(g'(x)\) into the quotient rule equation.

In the given exercise, we needed to differentiate the fraction \((x^2 + 1)/(x^2 - 1)\). We started by identifying \( f(x) = x^2 + 1\) and \(g(x) = x^2 - 1\). We then computed their derivatives, \(f'(x) = 2x\) and \(g'(x) = 2x\), respectively. Using the quotient rule, we got:
\[ v' = \frac{(2x)(x^{2} - 1) - (x^{2}+1)(2x)}{(x^{2} - 1)^2} \]
Simplifying this expression provides the derivative needed for further computation.
Derivative Calculation
In the final steps, we use the product and quotient rules to fully differentiate the function. First, identify the parts: \(u = x^3 - 8\) and \(v = \frac{x^2 + 1}{x^2 - 1}\).
We found \(u' = 3x^2\) and applying the quotient rule, worked out \(v' = \frac{(2x)(x^{2} - 1) - (x^{2}+1)(2x)}{(x^{2} - 1)^2}\).
Then, we combined the parts using the product rule:
\[ g'(x) = (3x^2) \times \frac{x^2 + 1}{x^2 - 1} + (x^3 - 8) \times v' \]
Finally, substitute \(v'\) back in and simplify the entire expression to find \(g'(x)\), the desired derivative.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Let \(f(x)=\frac{x^{2}}{x^{2}-1}\) and \(g(x)=\frac{1}{x^{2}-1}\) a) Compute \(f^{\prime}(x)\) b) Compute \(g^{\prime}(x)\) c) What can you conclude about the graphs of \(f\) and \(g\) on the basis of your results from parts (a) and (b)?

A proof of the Product Rule appears below. Provide a justification for each step. $$\text { a) } \frac{d}{d x}[f(x) \cdot g(x)]=\lim _{h \rightarrow 0} \frac{f(x+h) g(x+h)-f(x) g(x)}{h}$$ $$\text { b) } \quad=\lim _{h \rightarrow 0} \frac{f(x+h) g(x+h)-f(x+h) g(x)+f(x+h) g(x)-f(x) g(x)}{h}$$ $$\text { c) } \quad=\lim _{h \rightarrow 0} \frac{f(x+h) g(x+h)-f(x+h) g(x)}{h}+\lim _{h \rightarrow 0} \frac{f(x+h) g(x)-f(x) g(x)}{h}$$ $$\begin{array}{ll} \text { d) } & =\lim _{h \rightarrow 0}\left[f(x+h) \cdot \frac{g(x+h)-g(x)}{h}\right]+\lim _{h \rightarrow 0}\left[g(x) \cdot \frac{f(x+h)-f(x)}{h}\right] \end{array}$$ $$\text { e) } \quad=f(x) \cdot \lim _{h \rightarrow 0} \frac{g(x+h)-g(x)}{h}+g(x) \cdot \lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$$ $$\mathbf{f}) \quad=f(x) \cdot g^{\prime}(x)+g(x) \cdot f^{\prime}(x)$$ $$\text { g) } \quad=f(x) \cdot\left[\frac{d}{d x} g(x)\right]+g(x) \cdot\left[\frac{d}{d x} f(x)\right]$$

Find \(d y / d x .\) Each function can be differentiated using the rules developed in this section, but some algebra may be required beforehand. $$y=\frac{x^{5}+x}{x^{2}}$$

Sparkle Pottery has determined that the cost, in dollars, of producing \(x\) vases is given by \(C(x)=4300+2.1 x^{0.6}\) If the revenue from the sale of \(x\) vases is given by \(R(x)=65 x^{0.9},\) find the rate at which the average profit per vase is changing when 50 vases have been made and sold.

For each of the following, graph \(f\) and \(f^{\prime}\) and then determine \(f^{\prime}(1) .\) For Exercises use Deriv on the \(T I-83\). $$f(x)=\frac{5 x^{2}+8 x-3}{3 x^{2}+2}$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.