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Chapter 2: Q. 95 (page 236)

Prove each of the differentiation formulas in Exercises 93鈥�96. (These exercises involve hyperbolic functions.)

ddxcosh-1x=1x2-1.

Short Answer

Expert verified

We proved the formuladdxcosh-1x=1x2-1.

Step by step solution

01

Step 1. Given Information 

We need to prove the hyperbolic function ddxcosh-1x=1x2-1.

02

Step 2. Proving the function  

coshcosh-1x=x鈬�ddxcoshcosh-1x=ddxx鈬�sinhcosh-1xddxcosh-1x=1鈬�ddxcosh-1x=1sinhcosh-1x鈬�ddxcosh-1x=1cosh2cosh-1x-1鈬�ddxcosh-1x=1x2-1

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Most popular questions from this chapter

Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

f(x)=3x-4(2x+1)6

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鈥檚 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?

Use (a) the h鈫�0definition of the derivative and then

(b) the z鈫�cdefinition of the derivative to find f'(c) for each function f and value x=c in Exercises 23鈥�38.

25.f(x)=1x,x=-1

Use (a) the h鈫�0definition of the derivative and then

(b) the z鈫�cdefinition of the derivative to find f'(c) for each function f and value x=c in Exercises 23鈥�38.

26.f(x)=1x2,x=2

Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

f(x)=1x-3x2x5-1x
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