The Basic Derivative

The first topic we will cover is the derivative, and we will start on the most basic level.  Generally speaking, most Calculus 1 classes will begin their material discussing the derivative.  Lets first discuss how to find the derivative.  If you are given a function, lets say axⁿ, the derivative of this function would be

(n*a)x^n-1.

In words, you are multiplying the exponent by the coefficient.  Then, you subtract 1 from the exponent.  For example, if you are given the following function:

f(x) = 3x²
f '(x) = (3*2)x^2-1
f '(x) = 6x¹
f '(x) = 6x

This process is called differentiation.  Differentiation is very useful in applied mathematics.  More or less, the derivative of a function will give you the slope at whichever x-value you plug in.  It is something that never goes away as long as you are studying calculus!

There are a couple of easy, special cases to be aware of.  Given the function:

f(x) = 2x

, the derivative is quite simple.

f '(x) = (1*2)x^1-1
f '(x) = 2x⁰
f '(x) = 2(1)
f '(x) = 2

As you can see here, the exponent subtracts to zero.  Using the order of operations, we see that the x⁰ is executed first  Therefore, anything raised to the power of 0 is 1.  This gives us our constant times 1.  So just an easy rule to remember, the derivative of ax will always be a, a being any constant.  Also, the derivative of a constant, 5 for example, will always be 0.  This covers the very basics of the derivative.  In the next post, we'll discuss differentiation of a basic equation acting as a function.  Feel free to leave comments or ask questions.