Product Rule

When a function consists of a function multiplied with another function, the derivative of this new function cannot be calculated with only using the standard derivation rules. For this case another rule called the Product Rule must be used. The Product Rule can be written as shown below:

Let a function $h(x)$ consist of the product of the 2 functions $f(x)$ and $g(x)$:

The derivative $h'(x)$ is then equal to:

The derivative of a function that consists of the product of 2 functions $(h'(x))$ equals the product of the derivative of the first function $(f'(x))$ and the second function $(g(x))$ plus the first function $(f(x))$ times the derivative of the second function $(g'(x))$.

Example

The derivative $h'(x)$ of the function $h(x)$ shown below must be calculated. It can be easily seen that the function consists of 2 functions multiplied together.

In this case the function $h(x)$ consists of the functions $f(x) = x$ and $g(x) = sin(x)$ thus the Product Rule must be applied. The working out of the Product Rule is shown below.

1. $f(x) = x$ and $g(x) = sin(x)$ (The different parts of the Product Rule are identified)
2. $h'(x) = 1 * sin(x) + xcos(x)$ (The Product rule is used with the standard derivation rules)
3. $h'(x) = sin(x) + xcos(x)$ (The answer is written neatly)

Now is shown a method for solving derivatives of functions that consist of a product of 2 functions and an example derivative is worked out. In the next article the Qoutient rule will be explained. A rule necessary for calculating derivatives of a function that consists of one function divided by another function.