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.

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