   Chapter 3.11, Problem 29E

Chapter
Section
Textbook Problem

Prove the formulas given in Table 6 for the derivatives of the following functions.(a) cosh–1(b) tanh–1(c) csch–1(d) sech–1(e) coth–1

(a)

To determine

To prove: The formula for derivative of the function, ddx(cosh1x)=1x21.

Explanation

Proof:

Let y=cosh1x.

Implies that coshy=x and y0.

coshy=x

Differentiate with respect to x,

ddx(coshy)=ddx(x)sinhydydx=1                (using chain rule)dydx=1sinhy

Since the identity cosh2ysinh2

(b)

To determine

To prove: The formula for derivative of the function, ddx(tanh1x)=11x2.

(c)

To determine

To prove: The formula for derivative of the function, ddx(csch1x)=1|x|1+x2.

(d)

To determine

To Prove: The formula for derivative of the function, ddx(sech1x)=1x1x2.

(e)

To determine

To Prove: The formula for derivative of the function, coth1x=11x2.

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