Second Derivatives and Beyond!

Now that we have been introduced to the basic idea of derivatives, we will discuss multiple differentiation.  If you are given a function and asked to find the second derivative, you do just that.  Lets say this is your function:

f(x) = 3x⁴ - 5x² + 17x

Now, the first thing you will do is differentiate this function.  So, following the guidelines we have discussed, that would look something like this:

f(x) = 3x⁴ - 5x² + 17x
f '(x) = (4*3)x^4-1 - (2*5)x^2-1 + (1*17)^1-1
f '(x) = 12x³ + 10x + 17

Easy enough, right? Now that you have found your first derivative, we can move on the the second.  A second derivative is generally noted by two apostrophe characters.  In our example, it will be f ''(x), or "F double-prime of x" in words.  To find the second derivative, you simply differentiate your first derivative.  So lets continue from our original function:

f '(x) = 12x³ + 10x + 17
f ''(x) = (3*12)x^3-1 + (1*10)x^1-1 + 0
f ''(x) = 36x² + 10

And there we have it, our second derivative.  This concept is fairly easy to follow.  If you wanted to find the third derivative, you would follow the same steps.  Take your second derivative, and differentiate.  Just to prove the point, we'll continue with the function:

f ''(x) = 36x² + 10
f '''(x) = (2*36)x^2-1 + 0
f '''(x) = 72x

We now have our third derivative.  You could go on and on, as far as necessary, repeating the same steps to find further derivatives.  In our case, anything beyond the third derivative in a function will not be necessary.  So, I hope this post left you understanding how to find multiple derivatives of the same function.  Post for any further questioning.