   Chapter 3.3, Problem 17E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Prove that d d x ( csc x ) = − csc x cot x .

To determine

To prove: The derivative of csc x is cscxcotx. That is, ddx(cscx)=cscxcotx.

Explanation

Formula used:

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)

Proof:

Consider ddx(cscx).

Since cscx=1sinx, ddx(cscx)=ddx(1sinx).

Apply the quotient rule as shown in equation (1).

Substitute 1 for f(x) and sinx for g(x) in equation (1)

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