Qoutient Rule

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

Let a function $h(x)$ consist of the qoutient 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 qoutient of 2 functions $(h'(x))$ equals the product of the derivative of the first function $(f'(x))$ and the second function $(g(x))$ minus the first function $(f(x))$ times the derivative of the second function $(g'(x))$ which in total is then divided by the function $(g(x)^2)$ .

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 a function divided by another function.

In this case the function $h(x)$ consists of the functions $f(x) = ln(x)$ and $g(x) = x^3$ thus the Qoutient Rule must be applied. The working out of the Qoutient Rule is shown below.

1. $f(x) = ln(x)$ and $g(x) = x^3$ (The different parts of the Qoutient Rule are identified)
2. $h'(x) = \frac{\frac{1}{x}*x^3 - ln(x)* 3x^2}{(x^3)^2}$ (The Qoutient rule is used with the standard derivation rules)
3. $h'(x) = \frac{x^2 - 3x^2ln(x)}{x^6}$ (The answer is written neatly)

Now is shown a method for solving derivatives of functions that consist of a qoutient of 2 functions and an example derivative is worked out. In the next article the Chain Rule will be explained in more detail.