# Derivatives

One of the most important calculus concepts to learn is derivation.  Here I will show you how to calculate some simple derivatives.  The derivative looks like this $\frac{\text{d}}{\text{d}x}$ and can be read as “derivative with respect to x” so if you have a function $y = 5x$ and you are supposed to perform $\frac{\text{d}f}{\text{d}x}$ then you are going to be taking the derivative of the function “f” with respect to all variables “x”.

If n is a positive integer, this is your theorem:

$\frac{\text{d}}{\text{d}x} x^{n} = n x^{n-1}$

let’s apply this to our equation $y = 5x$.  Our derivative is looks like this to start $\frac{\text{d}f}{\text{d}x}(5x)$ where n = 1 because 5x is not raised to any power.

The first derivative will be represented like so:

$\frac{\text{d}f}{\text{d}x} = 5$

or

$f^{I} = 5$

Our equation is now equal to a constant.  If you take the derivative of ANY constant, the result is Zero.  So here $f^{II} = 0$

Let’s make this example a little more realistic, our new equation is $y = 8 x^{2} + 4x + 3$

$f^{I} = 16 x + 4$ each piece of the equation loses 1 order ($x^{2}$ second order), keeping in mind that the 3 was a constant and is now zero.

we can continue all the way to zero.

$f^{II} = 16$ $f^{III} = 0$