# Derivative of the Difference of Functions

It is given that the derivative of a function that is the difference of two other functions, is equal to the difference of their derivatives. This can be proved by using derivative by definition or the first principle method.

Consider a function of the form $y = f\left( x \right) - g\left( x \right)$.

First we take the increment or small change in the function:

Putting the value of function $y = f\left( x \right) - g\left( x \right)$ in the above equation, we get

Dividing both sides by $\Delta x$, we get

Taking the limit of both sides as $\Delta x \to 0$, we have

By the definition of derivative we have

This shows that the derivative of the difference of the two given functions is equal to the difference of their derivatives.

This difference rule can be expand to more than two function as

Example: Find the derivative of $y = \left( {4{x^3}} \right) - \left( {6x + 5} \right)$

We have the given function as

Differentiating with respect to variable $x$, we get

Now using the formula derivatives, we have