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

Find the derivatives of each of the functions in Exercises 17–50. In some cases it may be convenient to do some preliminary algebra.

f(x)=log33xsin2(x)+cos2(x).

Short Answer

Expert verified

The derivative of the given function is: 1.

Step by step solution

01

Step 1. Given Information.

f(x)=log33xsin2(x)+cos2(x).

02

Step 2. General formulas for finding derivatives.  

ddxxn=nxn-1,ddxsinx=cosx,ddxcosx=-sinx,ddxtanx=sec2x,ddx(secx)=secxtanx,ddx(cscx)=-cscxcotx,ddx(cotx)=-csc2x,Productrule:ddx(uv)=udvdx+vdudx,Quotientrule:ddxuv=vdudx-udvdxv2,Chainrule:ddxfog(x)=ddxf(g(x))×ddx(g(x).

03

Step 3. Solving derivative using above formulas. 

Solvingthefunctionbeforedifferentiating,weget,f(x)=xlog33sin2(x)+cos2(x),f(x)=x1=x.Since,log33=1andalsosin2(x)+cos2(x)=1.Nowdifferentiatingweget,f(x)=x,ddx(f(x))=1.

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

If Katie walked at 3miles per hour for 20minutes and then sprinted at 10miles an hour for 8minutes, how fast would Dave have to walk or run to go the same distance as Katie did at the same time while moving at a constant speed? Sketch a graph of Katie’s position over time and a graph of Dave’s position over time on the same set of axes.

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

For each function f(x)and interval a,bin Exercises 81-86, use the Intermediate Value Theorem to argue that the function must have at least one real root on a,b. Then apply Newton’s method to approximate that root.

f(x)=x3+1,a,b=-2,1

The following reciprocal rules tells us hoe to differentiate the reciprocal of a function

ddx(1f(x))=-1[f(x)]2

Prove this using

a) definition of the derivative

b) by using the quotient rule

The total yearly expenditures by public colleges and universities from 1990 to 2000 can be modeled by the function E(t)=123(1.025)t, where expenditures are measured in billions of dollars and time is measured in years since 1990.

(a) Estimate the total yearly expenditures by these colleges and universities in 1995.

(b) Compute the average rate of change in yearly expenditures between 1990 and 2000.

(c) Compute the average rate of change in yearly expenditures between 1995 and 1996.

(d) Estimate the rate at which yearly expenditures of public colleges and universities were increasing in 1995.

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