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Differentiate each of the functions in Exercises 29–34 in two

different ways: first with the product and/or quotient rules and

then without these rules. Then use algebra to show that your

answers are the same.

f(x)=xx-2x3

Short Answer

Expert verified

The derivative of the given function is-12x-32

Step by step solution

01

Step1. Given information  

given function is f(x)=xx-2x3

We need to differentiate first with the quotient rule and then without these rules, after that, we have to use algebra to show that the answers are the same.

02

Step 2. Differentiate using the quotient rule 

The quotient rule states that

Iffandgarefunctionsandbothfandgaredifferentiable,thenquotientrulefg'(x)=f(x)'g(x)f(x)g(x)'(g(x))2

localid="1649046997566" letf(x)=xg(x)=x-2x3Whenweapplyquotientrule,wegetderivativefunctionasddx(xx-2x3)=d(x)dx×(x-2x3)-xd(x-2x3)dx(x-2x3)2=(ddx(x12)(x-2x3)-x×ddx(x-2x3)(x-2x3)2sincex=x12=(ddx(x12)(x)-x×ddx(x)(x)2weknowthatxm×xn=xm+nsox-2×x3=x1andx1=x=12x×x-12-x(x)2sincepowerruleddx(xn)=nxn-1soddx(x12)=12x-12andddx(x)=1

03

Step 3. Differentiate with out using quotient  rule 

given function is

f(x)=xx-2x3wecanwritex=x12,byusinglawsofexponentx

04

Step 4. Checking  both the answers are the same

Without using the quotient rule derivative of the given function is

ddx(xx-2x3)=-12x-32thenbyapplyingthequotientrulederivativesofthegivenunctionisddx(xx-2x3)=12x×x-12-x12(x)2applythelawofexponentsandsolvetheexpressionbecomes=12x-12+1-x12x-2since1xm=x-mandxmxn=xm+n=12x-12+1-2-x12-2=12x-32-x-32=-12x-32sobyusingthealgebrabothanswersaresame.

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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.

Last night Phil went jogging along Main Street. His distance from the post office t minutes after 6:00p.m. is shown in the preceding graph at the right.

(a) Give a narrative (that matches the graph) of what Phil did on his jog.

(b) Sketch a graph that represents Phil’s instantaneous velocity t minutes after 6:00p.m. Make sure you label the tick marks on the vertical axis as accurately as you can.

(c) When was Phil jogging the fastest? The slowest? When was he the farthest away from the post office? The closest to the post office?

Think about what you did today and how far north you were from your house or dorm throughout the day. Sketch a graph that represents your distance north from your house or dorm over the course of the day, and explain how the graph reflects what you did today. Then sketch a graph of your velocity.

Use (a) the h0definition of the derivative and then

(b) the zcdefinition of the derivative to find f'(c) for each function f and value x=c in Exercises 23–38.

24.f(x)=x3,x=1

Suppose h(t) represents the average height, in feet, of a person who is t years old.

(a) In real-world terms, what does h(12) represent and what are its units? What does h' (12) represent, and what are its units?

(b) Is h(12) positive or negative, and why? Is h'(12) positive or negative, and why?

(c) At approximately what value of t would h(t) have a maximum, and why? At approximately what value of t would h' (t) have a maximum, and why?

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