Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

In the text we noted that if f(u(v(x))) was a composition of three functions, then its derivative is dfdx=dfdu×dudv×dvdx. Write this rule in “prime” notation.

Short Answer

Expert verified

In prime notation the derivative is:

fuv'(x)=f'(uv)(x)×(uv)'(x)×v'(x)

Step by step solution

01

Step 1. Given information:

The composite function and its derivative are:

f(u(v(x)))

dfdx=dfdu×dudv×dvdx

02

Step 2. Derivative in the prime notation:

The derivative of the given composite function is written as:

fuv'(x)=f'(uv)(x)×(uv)'(x)×v'(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 Vaia!

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

Find the derivatives of the functions in Exercises 21–46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

f(x)=53x4-13+3x-1100

Find the derivatives of the functions in Exercises 21–46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

f(x)=3x-4(2x+1)6

Prove, in two ways, that the power rule holds for negative integer powers

a) by using the zxdefinition of the derivative

b) by using theh0definition of the derivative

Every morning Linda takes a thirty-minute jog in Central Park. Suppose her distance s in feet from the oak tree on the north side of the park tminutes after she begins her jog is given by the function s(t)shown that follows at the left, and suppose she jogs on a straight path leading into the park from the oak tree.

(a) What was the average rate of change of Linda’s distance from the oak tree over the entire thirty-minute jog? What does this mean in real-world terms?

(b) On which ten-minute interval was the average rate of change of Linda’s distance from the oak tree the greatest: the first 10minutes, the second 10minutes, or the last10minutes?

(c) Use the graph of s(t)to estimate Linda’s average velocity during the 5-minute interval fromt=5tot=10. What does the sign of this average velocity tell you in real-world terms?

(d) Approximate the times at which Linda’s (instantaneous) velocity was equal to zero. What is the physical significance of these times?

(e) Approximate the time intervals during Linda’s jog that her (instantaneous) velocity was negative. What does a negative velocity mean in terms of this physical example?

Use the differentiation rules developed in this section to find

the derivatives of the functions

f(x)=4-x7

See all solutions

Recommended explanations on Math Textbooks

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.

Sign-up for free