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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)=x(x-1+1)

Short Answer

Expert verified

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

Step by step solution

01

Step1. Given information 

Here f(x)=x(x-1+1)

We have to differentiate with the product rule and

then without rule, after that, use algebra to show that the answers are the same.

02

Step 2. Differentiate using the product  rule 

When we apply the product rule on the given function , then TheProductRule:(f'g)(x)=f'(x)g(x)+f(x)g'(x)heretakef(x)=x,wecanrewritex=x12g(x)=x-1+1,Thenf'(x)=12x-12,sincepowerrule:fornd(xn)dx=nxn-1g'(x)=-1×x-1-1,constantruleandpowerruleapplied=-x-2nowapplytheproductrulewegetd(x(x-1+1dx=12x-12×x-1+1+x×-x-2=12x-12(x-1+1)-x12x-2

03

Step 3. Differentiate with out using product rule 

Here function is

f(x)=x(x-1+1)Wecanwritexasx12andapplydistributionlawonthegivenfunctionf(x)=x12x-1+x12,weknowthatxmxn=xm+n,then=x-12+x12nowderivativesofthefunctiond(f(x))dx=d(x-12+x12)dx=d(x-12)dx+d(x12)dxsumruleappliedhere=-12x-12-1+12x12-1sincepowerruled(xn)dx=nxn-1=-12x-32+12x-12

04

Step 4. Checking  both the answers are the same

By using product rule derivatives of the function is

d(x(x-1+1dx=12x-12(x-1+1)-x12x-2,firstusingthedistributivepropertywecanwearrangeandthensolve,thenweget=12x-12x-1+12x-12-x12x-2=12x-32+12x-12-x-32sincexmxn=xm+n=-12x-32+12x-12

without applying the product rule derivatives of the function is

d(x(x-1+1dx=-12x-32+12x-12

Here using the algebra both answers are same.

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