/*! 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 17 Use Version I of the Chain Rule ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 17

Use Version I of the Chain Rule to calculate \(\frac{d y}{d x}\). $$y=\sec e^{x}$$

Short Answer

Expert verified
Question: Determine the derivative of the function y = sec(e^x) with respect to x, using the Chain Rule. Answer: The derivative of the function y = sec(e^x) with respect to x is given by: dy/dx = sec(e^x)tan(e^x)e^x.

Step by step solution

01

Identify the outer function and the inner function

In our function, y = sec(e^x), the outer function \(f(x)\) is the secant function and the inner function \(g(x)\) is the exponential function: $$f(x) = \sec(x)$$ $$g(x) = e^x$$
02

Calculate the derivatives of the outer and inner functions

Next, we need to find the derivatives of the outer and inner functions. For the outer function \(f(x) = \sec(x)\), the derivative is: $$f'(x) = \sec(x)\tan(x)$$ For the inner function \(g(x) = e^x\), the derivative is: $$g'(x) = e^x$$
03

Apply the Chain Rule

Now, we can apply the Chain Rule to compute the derivative of y with respect to x: $$\frac{dy}{dx} = f'(g(x))g'(x) = \sec(g(x))\tan(g(x))g'(x)$$
04

Substitute the expressions for g(x) and g'(x)

Replace \(g(x)\) and \(g'(x)\) with their expressions for sec(e^x): $$\frac{dy}{dx} = \sec(e^x)\tan(e^x)e^x$$ Thus, the derivative of the function y = sec(e^x) with respect to x is: $$\frac{dy}{dx} = \sec(e^x)\tan(e^x)e^x$$

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

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.)

A particle travels clockwise on a circular path of diameter \(R,\) monitored by a sensor on the circle at point \(P ;\) the other endpoint of the diameter on which the sensor lies is \(Q\) (see figure). Let \(\theta\) be the angle between the diameter \(P Q\) and the line from the sensor to the particle. Let \(c\) be the length of the chord from the particle's position to \(Q\) a. Calculate \(d \theta / d c\) b. Evaluate \(\left.\frac{d \theta}{d c}\right|_{c=0}\)

Calculate the following derivatives using the Product Rule. $$\begin{array}{lll} \text { a. } \frac{d}{d x}\left(\sin ^{2} x\right) & \text { b. } \frac{d}{d x}\left(\sin ^{3} x\right) & \text { c. } \frac{d}{d x}\left(\sin ^{4} x\right) \end{array}$$ d. Based upon your answers to parts (a)-(c), make a conjecture about \(\frac{d}{d x}\left(\sin ^{n} x\right),\) where \(n\) is a positive integer. Then prove the result by induction.

Find \(f^{\prime}(x), f^{\prime \prime}(x),\) and \(f^{\prime \prime \prime}(x)\) \(f(x)=\frac{x^{2}-7 x}{x+1}\)

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. $$f(x)=x^{2 / 3}, \text { for } x>0$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

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.