Differentiation of Power Functions and Exponentials

Power Functions

A power function is when we have any variable x to the power of any real number represented here as n.


In the formula sheet we are given the equation:


So basically all we do is multiply our function by the power term and then minus one the power term by one.

Power Function Differentiation Example

Say we have x to the power of negative fraction:


We still follow the same procedure of multiplying the function by the power term and then minus it by one.

\frac{dy}{dx}=-\frac{5}{2}\times x^{-\frac{5}{2}-1}

\frac{dy}{dx}=-\frac{5}{2} x^{-\frac{7}{2}}


A polynomial is basically a combination of power functions. So while they may appear to be more tricky, if we isolate each term, they are just as easy as the example above.

Polynomial Differentiation example:

Given the equation below find the derivative:


you first need to rewrite it so that it looks more straightforward:


Then just follow the simple rule: bring the power down, put it in front, and take one away from the power.

Lets start with  first term:

3*x^{2}\newline 2\times 3x^{2-1}\newline -3x

The second term:

3x^{-1}\newline -1\times 3x^{-1-1}\newline -3x^{-2}

The plus 4 at the end of equation is eliminated, as it has no x component.

To find \frac{dy}{dx} we simply add these terms together

\frac{dy}{dx}=6x-3x^{-2}+0\newline =6x-3x^{-2}

Exponential Functions

The exponential functions that we focus on in maths methods is e^{x}

This is Euler’s number to the power of a variable x and has a remarkable differentiation property

Exponential Differentiation

In the formula sheet we are provided with the very useful formula:


This is one of the easiest rules to follow as all you do is multiply the exponential component by the coefficient of x

Exponential Differentiation Example

Let’s try to find the derivative for this function

y=e^{3x}\newline \frac{dy}{dx}=3 \times e^{3x} \newline =3e^{3x}   

If we follow the formula provided, it greatly simplifies the process