Find derivative, and explain step by step. 21. y = csc ³ (cos 8x)

Answered: 21. y = csc³ (cos 8x)

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

See similar textbooks

Related questions

Question

Find derivative, and explain step by step.

Expert Solution

Step 1: Introduction

Given,

$y equals text csc end text cubed space open parentheses cos open parentheses 8 x close parentheses close parentheses$

The aim is to find the derivative of the above function.

Concept used :

The chain rule states that the derivative of $f left parenthesis g left parenthesis x right parenthesis right parenthesis$ is $fraction numerator d over denominator d x end fraction open square brackets f open curly brackets g left parenthesis x right parenthesis close curly brackets close square brackets times fraction numerator d over denominator d x end fraction left parenthesis g left parenthesis x right parenthesis right parenthesis$ .

The formula used here:

1. Derivative of $text csc end text left parenthesis x right parenthesis$ with respect to $x$ ,

$fraction numerator d over denominator d x end fraction left parenthesis text csc end text space x right parenthesis equals negative text cot end text space x times text csc end text space x$

2. Derivative of $cos left parenthesis x right parenthesis$ with respect to $x$ ,

$fraction numerator d over denominator d x end fraction left parenthesis cos space x right parenthesis equals negative sin left parenthesis x right parenthesis$

3. Derivative of $x to the power of n$ with respect to $x$

$fraction numerator d over denominator d x end fraction left parenthesis x to the power of n right parenthesis equals n x to the power of n minus 1 end exponent$