## Product Rule and Quotient Rule

# Product Rule and Quotient Rule

### Objectives

- State the constant, constant multiple, and power rules.
- Apply the sum and difference rules to combine derivatives.
- Use the product rule for finding the derivative of a product of functions.
- Use the quotient rule for finding the derivative of a quotient of functions.
- Extend the power rule to functions with negative exponents.
- Combine the differentiation rules to find the derivative of a polynomial or rational function.

### Summary

If a function is a sum, product, or quotient of simpler functions, then we can use the sum, product, or quotient rules to differentiate it in terms of the simpler functions and their derivatives.

The product rule tells us that if \(P\) is a product of differentiable functions \(f\) and \(g\) according to the rule \(P(x) = f(x) g(x)\text{,}\) then

\begin{equation*} P'(x) = f(x)g'(x) + g(x)f'(x)\text{.} \end{equation*}The quotient rule tells us that if \(Q\) is a quotient of differentiable functions \(f\) and \(g\) according to the rule \(Q(x) = \frac{f(x)}{g(x)}\text{,}\) then

\begin{equation*} Q'(x) = \frac{g(x)f'(x) - f(x)g'(x)}{g(x)^2}\text{.} \end{equation*}Along with the constant multiple and sum rules, the product and quotient rules enable us to compute the derivative of any function that consists of sums, constant multiples, products, and quotients of basic functions. For instance, if \(F\) has the form

\begin{equation*} F(x) = \frac{2a(x) - 5b(x)}{c(x) \cdot d(x)}\text{,} \end{equation*}then \(F\) is a quotient, in which the numerator is a sum of constant multiples and the denominator is a product. This, the derivative of \(F\) can be found by applying the quotient rule and then using the sum and constant multiple rules to differentiate the numerator and the product rule to differentiate the denominator.