   Chapter 8.4, Problem 73E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Think About It Rewrite the trigonometric function in terms of sine and/or cosine and then differentiate to prove the following differentiation rules.(a) d d x [ tan x ] = sec 2 x (b) d d x [ sec x ] = sec x  tan  x (c) d d x [ cot x ] = − csc 2 x (d) d d x [ csc x ] = − csc x   cot x

(a)

To determine

To prove: That ddx[tanx]=sec2x after writing trigonometric function in terms of Sine and/or Cosine.

Explanation

Given Information:

The trigonometric function:

ddx[tanx]=sec2x

Formula used:

Cosine differentiation rule:

ddx[sinu]=cosududx

Cosine differentiation rule:

ddx[cosu]=sinududx

Quotient rule of differentiation:

ddx[a(x)b(x)]=b(x)a(x)a(x)b(x)[b(x)]2

General power rule of differentiation:

ddx[xn]=nxn1

Formulas for trigonometric reciprocal identities:

cotθ=cosθsinθ,

sinθ=1cscθ

And,

sin2θ+cos2θ=1

Proof:

Consider the left side of trigonometric function:

ddx[tanx]

Use formula for reciprocal trigonometric identity and rewrite the function in terms of Sine and Cosine.

ddx[tanx]=ddx[sinxcosx]

Apply the quotient rule of differentiation to find the derivative of the above function

(b)

To determine

To prove: That ddx[secx]=sec2x after writing trigonometric function in terms of Sine and/or Cosine.

(c)

To determine

To prove: That ddx[cotx]=sec2x after writing trigonometric function in terms of Sine and/or Cosine.

(d)

To determine

To prove: That ddx[cscx]=cscxcotx after writing trigonometric function in terms of Sine and/or Cosine.

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