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Use the definition of the derivative to prove the following special case of the product rule

ddx(x2f(x))=2xf(x)+x2f'(x)

Short Answer

Expert verified

We proved the special case of product function using the definition of the derivative

Step by step solution

01

Given information

We are given a functionddx(x2f(x))=2xf(x)+x2f'(x)

02

Find the derivative

Consider g(x)=x2f(x)

Using the definition of derivative we get,

limh0g(x+h)-g(x)hlimh0(x+h)2f(x+h)-x2f(x)hlimh0(x2+2xh+h2)f(x+h)-x2f(x)hlimh0x2(f(x+h)-f(x))h+limh0h(2x+h)f(x+h)h=x2f'(x)+2xf(x)

Hence proved

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Most popular questions from this chapter

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.

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

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(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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role="math" localid="1648290170541" f(x)=x-1x+3

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(a) In real-world terms, what does T(5)represent and what are its units? What does T'(5)represent and what are its units?

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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.

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