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In Exercises 72–77, find a function f that has the given derivative f. In each case you can find the answer with an educated guess-and-check process. (Some of these exercises involve hyperbolic functions.)

f'(x)=2x1+4x2

Short Answer

Expert verified

The function that has the derivative f'(x)=2x1+4x2is f(x)=121+4x2.

Step by step solution

01

Step 1. Given Information.

The derivative:

f'(x)=2x1+4x2

02

Step 2. Guess the formula for the given derivative.

We know that,

ddx(sinh-1x)=11+x2

So we can conclude that the derivative will be of the form of hyperbolic sinefunction.

03

Step 3. Check guess-and-check process.

f(x)=sinh-1(2x)f'(x)=11+x2.(2)=21+x2

which is not a given derivative.

04

Step 4. Check another function.

Consider the function,

f(x)=121+4x2f'(x)=ddx(12(1+4x2))12f'(x)=12.[12(1+4x2)-12.8x]=12[4x1+4x2]=2x1+4x2

which is the given derivative.

So the function is f(x)=121+4x2

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