# The Power Rule of Derivatives

The Power Rule of derivatives is an essential formula in differential calculus. Now we shall prove this formula by definition or first principle.

Let us suppose that the function is of the form $y = f\left( x \right) = {x^n}$, where $n$ is any constant power.

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

Putting the value of function $y = {x^n}$ in the above equation, we get

Taking $x$ common from the above equation, we get

Now taking common ${x^n}$, we get

Expanding the above expression using binomial series, we get

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

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

This shows that the derivative of $x$ power $n$ is $n{x^{n - 1}}$.

Example: Find the derivative of $y = f\left( x \right) = 7{x^3} + 2$

We have the given function as

Differentiating with respect to variable $x$, we get

Now using the power rule and constant function rule, we have