Chapter 12.3, Problem 36E

To determine

To find: The derivative of the function.

Answer to Problem 36E

The derivative of the function is $-\frac{3}{x^4}$ .

Explanation of Solution

Given information:

The given function is,

  $f\left(x\right)=\frac{1}{x^3}$

Concept used:

The derivative of the function is calculated as,

  $f^{\prime }\left(x\right)=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}$

The given function is,

  $f\left(x\right)=\frac{1}{x^3}$

The derivative of the function is calculated as,

  $\begin{array}{c}{f}^{\prime }\left(x\right)=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}\\ =\underset{h\to 0}{\lim }\frac{\left[\frac{1}{{\left(x+h\right)}^3}\right]-\left[\frac{1}{x^3}\right]}{h}\\ =\underset{h\to 0}{\lim }\frac{\left[\frac{x^3-{\left(x+h\right)}^3}{x^3{\left(x+h\right)}^3}\right]}{h}\\ =-\frac{3}{x^4}\end{array}$

Therefore, the derivative of the functionis $-\frac{3}{x^4}$ .