   Chapter 9.5, Problem 38E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Use the Quotient Rule to show that the Powers of x Rule applies to negative integer powers. That is, show that ( d / d x ) x " = n x n − 1 when  n   =   − k , k >   0 , by finding the derivative of f ( x ) =   1   / ( x k ) .

To determine

To prove: The rule of a derivative ddx(xn)=nxn1 when n=k(k>0) using quotient rule by finding the derivative of f(x)=1xk.

Explanation

Given Information:

The provided function is,

f(x)=1xk

Formula used:

According to the quotient rule of derivative,

f(u(x)v(x))=v(x)(u(x))u(x)(v(x))[v(x)]2

Proof:

Consider the provided function,

f(x)=1xk

Now, use the quotient rule of derivative,

f(u(x)v(x))=v(x)(u(x))u(x)(v(x))[v(x)]2

To obtain the derivative of this function by substituting xk for v(x) and 1 for u(x) as,

f(x)=xk(<

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