Thursday, April 25, 2013

Differentiation Rules - The Power Rule

Welcome to my second post of my series on differentiation formulas.  So far in my recent posts, I have explored in depth all about the concept of derivatives and using differentiation to find them, and then I started this current series with an easy theorem to remember for finding the derivative of a constant function.  This follow-up post will now explain to you probably one of the most used methods for finding derivatives: The Power Rule for Differentiation.

First, allow me to present this rule to you in all it's nasty-looking glory:

The only condition that this rule has is that n must be a positive integer.  OK, maybe it's not as frightening as I would have made it out to be... but it still doesn't look very friendly.  However, when you start doing questions with it, you will understand the easy workflow process and find that it actually is very simple to remember and use!

Let's try a few.  Trust me, you will see the pattern and rhythm right away!

Find the derivative of f(x) = x3.

According to our rule, the n value here is 3, so then we have:

f'(x) = 3x3-1 = 3x2.

That's all there is to it!

Next, differentiate the function f(x) = x500.

This one seems a bit tougher, but you do the exact same thing.  Check it out:

f'(x) = 500x500-1 = 500x499

Easy, right?

Here's a simple image that can help you to remember what you have to do.  Like I said, once you start doing these, it is very hard to mess up.

Here's a better way to remember this, which may be even easier to think of but harder to draw:  imagine that the exponent is a pile of sequential number cards (e.g. 1, 2, 3, 4, 5...) with the high card on top.  When you have to find the derivative, take the top card and place it in front of the x, leaving the card beneath it (which is 1 number smaller) visible.  It's easier to actually do this than to draw it, but it's a great way to remember this formula because it is so easy to visualize!

For fun (!), you can see the proof of this by using the Binomial Theorem.  I won't go into that much detail here, but in a nutshell, if you start with the definition of a derivative and let f(x) = xn, you can expand the (x + h)n term with the Binomial Theorem.  Since the definition requires you to find the limit as h approaches zero, this causes all but the first term of the expansion to equal zero, leaving you with the above result.

So, now you know one of the most common differentiation rules.  In my next posts, I'll show you how to apply this rule to your terms when you have a constant value in front of the x.  It's as easy as to do as this one!  Thanks for reading, and make sure you click the Like and +1 buttons on this page!

Tuesday, April 23, 2013

Differentiation Rules - Derivative of a Constant Function

For those of you just tuning in, my last post was a mega-post about derivatives and an introduction to differential calculus.  If you need some help getting started with understanding how to find derivatives, I highly recommend giving that a read.  One impression you may have of this concept is that it requires a lot of work - lots of lengthy formulas and limit calculations.  While you could certainly use those methods for all of your differentiating questions, you would be wasting your time!  In my never-ending quest to show you mathematics made easy, today I am going to start a series of posts about differentiation formulas - essentially, shortcuts through a lot of the repetitive and lengthy work needed if you were to explicitly use the definition of a derivative!  Once you learn some of these tricks, you will fly through your calculus homework!

(To be honest, these aren't so much tricks as they are actual mathematical theorems and rules.)

The first rule should be simple to understand, if you think about it.  Take any constant function, f(x) = c.  The derivative of this, that is, f'(x), will always be zero.

It's an easy one to remember, and the explanation is easy to visualize as well.  If you have a graph of f(x) = c, you basically have a horizontal line that never varies.  For every value of x on your curve, f(x) is the same, constant value.  So, it has no slope - its slope is zero.  Furthermore, if you refer back to my graphical explanation of derivatives in my previous post, you will see that the derivative f'(x) is equal to the slope of f(x).  So, if we have a line that has no slope here, we can see how this rule comes together.  For fun, you can practice your limit notation and long-hand derivative calculations to prove that this is the case.  Refer back again to my last post and use the definition of a derivative.  Hint: in your calculation, f(x) = c, and f(x + h) = c.

This is one of the easiest differentiation formulas (if you can call it that) that you are going to encounter, so memorize this one, and get ready for something a little bit more challenging in my next post: the Power Rule.

Please remember to click the Like and +1 buttons on this page if it was helpful!  I really appreciate it!

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