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Find the derivatives of each of the functions in Exercises 51–62. In some cases it may be convenient to do some preliminary algebra. (These exercises involve hyperbolic functions and their inverses.)

f(x)=sinh-1x3

Short Answer

Expert verified

The derivative is3x2x6+1.

Step by step solution

01

Step 1. Given information.

The given function isf(x)=sinh-1x3.

02

Step 2. Preliminary Algebra.

We know,

ddxxn=nxn-1ddxsinh-1x=1x2+1

03

Step 3. Derivative.

The derivative of the function is,

ddxsinh-1x3=1x32+1ddxx3=1x6+13x2=3x2x6+1

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Most popular questions from this chapter

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?

Use the definition of the derivative to find f'for each function fin Exercises 34-59

f(x)=1x2

Stuart left his house at noon and walked north on Pine Street for 20minutes. At that point he realized he was late for an appointment at the dentist, whose office was located south of Stuart’s house on Pine Street; fearing he would be late, Stuart sprinted south on Pine Street, past his house, and on to the dentist’s office. When he got there, he found the office closed for lunch; he was 10minutes early for his 12:40appointment. Stuart waited at the office for 10minutes and then found out that his appointment was actually for the next day, so he walked back to his house. Sketch a graph that describes Stuart’s position over time. Then sketch a graph that describes Stuart’s velocity over time.

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)=1-4x2(3x2+1)9

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

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

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