/*! 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 58 Use the Chain Rule combined with... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 58

Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. $$y=\tan \left(x e^{x}\right)$$

Short Answer

Expert verified
Answer: The derivative of the function \(y = \tan(xe^x)\) with respect to x is \(\frac{dy}{dx} = \sec^2(xe^x) \cdot e^x(1 + x)\).

Step by step solution

01

Differentiate the outer function using the trigonometric differentiation rule

Let \(u=xe^x\). The given function can be rewritten as \(y=\tan(u)\). Now, we differentiate the function \(y\) with respect to \(u\), bearing in mind that the derivative of the tangent function is \(\sec^2(u)\): $$\frac{dy}{du} = \sec^2(u) = \sec^2(xe^x)$$
02

Differentiate the inner function using the Product Rule

We have \(u=xe^x\). To find the derivative of this function with respect to x, we will employ the Product Rule, which is given by: $$\frac{d(uv)}{dx} = u'\cdot v + u\cdot v' $$ In our case, \(u = x\), \(v = e^x\). Therefore, we have \(u' = 1\) and \(v' = e^x\): $$\frac{du}{dx} = x'(e^x) + x(e^x)' = 1\cdot e^x + x\cdot e^x = e^x(1 + x)$$
03

Apply the Chain Rule

We will now use the Chain Rule, which states that: $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$ Substituting the values we have found in Steps 1 and 2: $$\frac{dy}{dx} = \sec^2(xe^x) \cdot e^x(1 + x)$$
04

Simplify the result (if possible)

In this case, there isn't much simplification needed. The final derivative of the given function is: $$\frac{dy}{dx} = \sec^2(xe^x) \cdot e^x(1 + x)$$ The derivative of the function \(y = \tan(xe^x)\) with respect to x is \(\frac{dy}{dx} = \sec^2(xe^x) \cdot e^x(1 + x)\).

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

Cobb-Douglas production function The output of an economic system \(Q,\) subject to two inputs, such as labor \(L\) and capital \(K\) is often modeled by the Cobb- Douglas production function \(Q=c L^{a} K^{b} .\) When \(a+b=1,\) the case is called constant returns to scale. Suppose \(Q=1280, a=\frac{1}{3}, b=\frac{2}{3},\) and \(c=40\) a. Find the rate of change of capital with respect to labor, \(d K / d L\) b. Evaluate the derivative in part (a) with \(L=8\) and \(K=64\)

A port and a radar station are 2 mi apart on a straight shore running east and west. A ship leaves the port at noon traveling northeast at a rate of \(15 \mathrm{mi} / \mathrm{hr}\). If the ship maintains its speed and course, what is the rate of change of the tracking angle \(\theta\) between the shore and the line between the radar station and the ship at 12: 30 p.m.? (Hint: Use the Law of sines.)

Consider the following functions (on the given interval, if specified). Find the inverse function, express it as a function of \(x,\) and find the derivative of the inverse function. $$f(x)=\sqrt{x+2}, \text { for } x \geq-2$$

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

Graphing with inverse trigonometric functions a. Graph the function \(f(x)=\frac{\tan ^{-1} x}{x^{2}+1}\) b. Compute and graph \(f^{\prime}\) and determine (perhaps approximately) the points at which \(f^{\prime}(x)=0\) c. Verify that the zeros of \(f^{\prime}\) correspond to points at which \(f\) has a horizontal tangent line.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Calculus

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.