   Chapter 3.3, Problem 31E

Chapter
Section
Textbook Problem

(a) Use the Quotient Rule to differentiate the function f ( x ) = tan x − 1 sec x (b) Simplify the expression for f(x) by writing it in terms of sin x and cos x, and then find .f'(x).(c) Show that your answers to parts (a) and (b) are equivalent.

(a)

To determine

To find: The derivative of f(x)=tanx1secx.

Explanation

Given:

The function is f(x)=tanx1secx.

Derivative rule: Quotient rule

If f(x) and g(x) are differentiable functions, then the quotient rule is,

ddx[f(x)g(x)]=g(x)ddx[f(x)]f(x)ddx[g(x)][g(x)]2 (1)

Calculation:

Obtain the derivative of f(x).

Substitute tanx1 for f(x) and secx for g(x) in equation (1),

ddx[tanx1secx]=[secx]ddx[tanx1][tanx1]ddx[secx][secx]2 =(secx)(sec2x)(tanx1)(secx

(b)

To determine

To simplify: The expression of f(x) in terms of sinx and cosx. and compute the derivative of f(x).

(c)

To determine

To show: The answers of parts (a) and (b) are equivalent. That is,1+tanxsecx=cosx+sinx.

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