Chapter 3, Problem 1CCE

State the product rule and quotient rule.

To determine

The product rule and the quotient rule.

Answer to Problem 1CCE

Solution:

Product rule:

$\frac{d}{dx}\left[f(x)g(x)\right]=f(x)g\text{'}(x)+g(x)f\text{'}(x)$

Quotient rule:

$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{g(x)f\text{'}(x)-f(x)g\text{'}(x)}{{[g(x)]}^2}$

Explanation of Solution

Given information:

Instructions to describe the product rule and the quotient rule.

Explanation:

If $f(x)$ and $g(x)$ are both differentiable functions, then the derivative of the product is determined by the following rule.

Product rule:

$\frac{d}{dx}\left[f(x)g(x)\right]=f(x)g\text{'}(x)+g(x)f\text{'}(x)$

If $f(x)$ and $g(x)$ are both differentiable functions, then the derivative of $\frac{f(x)}{g(x)}$ is determined by the following rule.

Quotient rule:

$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{g(x)f\text{'}(x)-f(x)g\text{'}(x)}{{[g(x)]}^2}$