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

Use the differentiation rules developed in this section to find the derivatives of the functions in Exercises 35-64. Note that it may be necessary to do some preliminary algebra before differentiating.

f(x)=x2x25

Short Answer

Expert verified

The derivative of the function isf'(x)=85x35.

Step by step solution

01

Step 1. Given Information

The given function isf(x)=x2x25.

02

Step 2. Simplify the function

Simplify the function.

f(x)=x2x25=x2-25=x85

03

Step 3. Find the derivative

Apply the power rule of derivative, (xn)'=nxn-1.

f'(x)=85x85-1=85x35

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

Taking the limit: We have seen that if f is a smooth function, then f'(c)โ‰ˆf(c+h)-f(c)hThis approximation should get better as h gets closer to zero. In fact, in the next section we will define the derivative in terms of such a limit.

f'(c)=limhโ†’0f(c+h)-f(c)h.

Use the limit just defined to calculate the exact slope of the tangent line tof(x)=x2atx=4.

A bowling ball dropped from a height of 400feet will be s(t)=400-16t2feet from the ground after tseconds. Use a sequence of average velocities to estimate the instantaneous velocities described below:

When the bowling ball hits the ground, withh=-0.5,h=-0.2andh=-0.1

Use the limit you just found to calculate the exact slope of the tangent line to f(x)=x2at x=4. Obviously you should get the same final answer as you did earlier.

Use the definition of the derivative to find ffor each function fin Exercises 39-54.

f(x)=x2-1x2-x-2

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 Lindaโ€™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 Lindaโ€™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 Lindaโ€™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 Lindaโ€™s (instantaneous) velocity was equal to zero. What is the physical significance of these times?

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

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