/*! 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 26 Find \(d y / d x\) for the follo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 26

Find \(d y / d x\) for the following functions. $$y=\frac{\left(x^{2}-1\right) \sin x}{\sin x+1}$$

Short Answer

Expert verified
Question: Find the derivative of the function $$y=\frac{\left(x^{2}-1\right) \sin x}{\sin x+1}$$ with respect to x. Answer: The derivative of the given function with respect to x is $$\frac{dy}{dx}=\frac{(x^2-1)\cos{x} + 2x\sin^2{x} + 2x\sin{x}}{(\sin{x}+1)^2}$$.

Step by step solution

01

Recall the Quotient Rule and Chain Rule

The Quotient Rule states that for a function of the form \(y = \frac{u}{v}\), where \(u = u(x)\) and \(v = v(x)\), the derivative is given by: $$\frac{dy}{dx}=\frac{u'v-uv'}{v^2}$$ The Chain Rule states that if we have a composition of functions, say \(y(u(x))\), then the derivative is given by: $$\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$$ We will apply these rules to find the derivative of the given function.
02

Identify the functions u and v

In this situation, we have: $$u(x)=(x^{2}-1) \sin{(x)}$$ $$v(x)=\sin{(x)}+1$$
03

Compute the first derivatives of u and v

First, we find the derivative of \(u(x)\) using the Product Rule: $$\frac{d}{dx}\left[(x^{2}-1)\sin{x}\right]=(x^{2}-1)\frac{d}{dx}\left[\sin{x}\right]+\sin{x} \frac{d}{dx}\left[x^{2}-1\right]$$ $$\Rightarrow u'(x)=(x^{2}-1)\cdot \cos{x}+\sin{x} \cdot 2x$$ Next, we find the derivative of \(v(x)\): $$\frac{d}{dx}\left[\sin{x}+1\right]=\frac{d}{dx}[\sin{x}]+\frac{d}{dx}[1]$$ $$\Rightarrow v'(x)=\cos{x}$$
04

Apply the Quotient Rule using the derivatives found above

Now that we have \(u'(x)\) and \(v'(x)\), we can apply the Quotient Rule: $$\frac{dy}{dx}=\frac{u'v-uv'}{v^2}$$ $$\frac{dy}{dx}=\frac{((x^{2}-1)\cos{x}+2x\sin{x})(\sin{x}+1)-((x^{2}-1)\sin{x})(\cos{x})}{(\sin{x}+1)^2}$$
05

Simplify the expression

Finally, we simplify the expression: $$\frac{dy}{dx}=\frac{(x^2-1)\cos{x}\sin{x} + 2x\sin^2{x} + (x^2-1)\cos{x} + 2x\sin{x} -(x^2-1)\sin{x}\cos{x}}{(\sin{x}+1)^2}$$ $$\frac{dy}{dx}=\frac{(x^2-1)\cos{x} + 2x\sin^2{x} + 2x\sin{x}}{(\sin{x}+1)^2}$$ Thus, the derivative of the given function is: $$\frac{dy}{dx}=\frac{(x^2-1)\cos{x} + 2x\sin^2{x} + 2x\sin{x}}{(\sin{x}+1)^2}$$

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!

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

Surface area of a cone The lateral surface area of a cone of radius \(r\) and height \(h\) (the surface area excluding the base) is \(A=\pi r \sqrt{r^{2}+h^{2}}\) a. Find \(d r / d h\) for a cone with a lateral surface area of \(A=1500 \pi\) b. Evaluate this derivative when \(r=30\) and \(h=40\)

Visualizing tangent and normal lines a. Determine an equation of the tangent line and normal line at the given point \(\left(x_{0}, y_{0}\right)\) on the following curves. (See instructions for Exercises \(63-68 .)\) b. Graph the tangent and normal lines on the given graph. \(\left(x^{2}+y^{2}-2 x\right)^{2}=2\left(x^{2}+y^{2}\right);\) \(\left(x_{0}, y_{0}\right)=(2,2)\) (lima莽on of Pascal)

Find \(f^{\prime}(x), f^{\prime \prime}(x),\) and \(f^{\prime \prime \prime}(x)\) \(f(x)=x^{2} e^{3 x}\)

Use the following table to find the given derivatives. $$\begin{array}{llllll} x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 5 & 4 & 3 & 2 & 1 \\ f^{\prime}(x) & 3 & 5 & 2 & 1 & 4 \\ g(x) & 4 & 2 & 5 & 3 & 1 \\ g^{\prime}(x) & 2 & 4 & 3 & 1 & 5 \end{array}$$ $$\left.\frac{d}{d x}\left[\frac{f(x) g(x)}{x}\right]\right|_{x=4}$$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The derivative \(\frac{d}{d x}\left(e^{5}\right)\) equals \(5 \cdot e^{4}\) b. The Quotient Rule must be used to evaluate \(\frac{d}{d x}\left(\frac{x^{2}+3 x+2}{x}\right)\) c. \(\frac{d}{d x}\left(\frac{1}{x^{5}}\right)=\frac{1}{5 x^{4}}\) d. \(\frac{d^{n}}{d x^{n}}\left(e^{3 x}\right)=3^{n} \cdot e^{3 x},\) for any integer \(n \geq 1\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Geometry

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.