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


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

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