   Chapter 3.4, Problem 95E

Chapter
Section
Textbook Problem

(a) If n is a positive integer, prove that d d x ( sin n x cos n x ) = n sin n − 1 x cos ( n + 1 ) x (b) Find a formula for the derivative of y = cosnx cos nx that is similar to the one in part (a).

(a)

To determine

To prove: If n is a positive integer, then ddx(sinnxcosnx)=nsinn1xcos(n+1)x .

Explanation

Result used: The Power Rule combined with the Chain Rule

If n is any real number and g(r) is differentiable function, then

ddr[g(r)]n=n[g(r)]n1g(r) (1)

Derivative Rule:

(1) Power Rule: ddx(xn)=nxn1.

(2) Product Rule: ddx[f(x)g(x)]=f(x)ddx[g(x)]+g(x)ddx[f(x)]

Proof:

Obtain the derivative ddx(sinnxcosnx).

Apply the product rule (2),

ddx(sinnxcosnx)=sinnxddx(cosnx)+cosnxddx(sinnx)=sinnx(sinnxn)+cosnxddx(sinnx)

=nsinnx(sinnx)+cosnxddx(sinnx) (2)

Obtain the derivative ddx(sinnx) by using the power rule combined with the chain rule as shown equation (1),

ddx(sinnx)=n(sinx)

(b)

To determine

To find: The derivative of y=cosnxcosnx that is similar to part (a).

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

In Exercises 1-6, simplify the expression. 3. 12t2+12t+34t21

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

The graph at the right has equation:

Study Guide for Stewart's Multivariable Calculus, 8th

What does Theorem 1, page 532, allow us to do?

Finite Mathematics for the Managerial, Life, and Social Sciences 