   Chapter 8.4, Problem 66E ### 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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# Finding Second Derivatives In Exercises 63–66, find the second derivative of the trigonometric function. y = csc 2   π x

To determine

To calculate: The second derivative of the trigonometric function y=csc2πx.

Explanation

Given Information:

The provided trigonometric function is y=csc2πx.

Formula used:

Cosecant differentiation rule:

ddx[cscu]=cscucotududx

Cotangent differentiation rule:

ddx[cotu]=csc2ududx

Product rule of differentiation:

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

General power rule of differentiation:

ddx[xn]=nxn1

Calculation:

Consider the trigonometric function:

y=csc2πx

Apply the general power rule and Cosecant rule of differentiation.

y=ddx[csc2πx]=2cscπx(πcscπxcotπx)=2πcsc2πxcotπx

Now, apply the product rule, general power rule, and Cosecant and Cotangent rule of differentiation on the above function again

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