Saturday, May 4, 2013

Differentiation Rules - Finding the Derivative of a Constant Times a Function

In this post I'm going to explain another one of the differentiation rules for working with derivatives.  This time, I will show you how to find the derivative of a constant times a function.

In case you have missed them, I am creating a series of posts that explain some basic concepts in differential calculus.  So far, in my first lesson I explained how to find the derivative of a constant function, and then I followed that with a post about the power rule for derivatives.  So far, these are some of the basic rules for calculating derivatives, and it is a good idea to become very familiar with them so that you can apply them all as required on more elaborate problems later on.

This derivatives rule is a very simple one, but that doesn't make it any less important.  Consider that you have any function f(x), and it is multiplied by a constant.  Assuming that f'(x) in fact exists, f'(x) times a constant is equal to the constant times this derivative.  That may sound a little wordy, so maybe this equation will be a little clearer:

So, all you really need to be concerned with is calculating the derivative f'(x).  Then, multiplying by the constant is a simple calculation that you can add at the end.

If you're curious about what this rule means, here is one way of looking at this.  If you have an equation such as y = f(x), if you consider multiplying this function by a constant, what you are doing is essentially stretching the graph vertically.  So, comparatively, y = 2f(x) is a graph that is twice the height.  Therefore, if you then consider the slope along this function, since you have doubled the rise but not the run (the slope calculation), you have a doubled slope at every point.  And since the derivative of a function represents the slope of the line at a point, you can then see how this rule all comes together.  Basically, if you stretch the function by a constant factor, you can simply multiply the slope (the derivative) by this factor as well.

As I said, there isn't much to remember about this particular derivative rule, but it is very important to know.  It will often need to be considered in addition to other rules that I have/will outline in this series. One example would be to calculate the derivative of something like 4x3.  In this case, you'd need to draw upon this rule, as well as the power rule from my last post.  There are a few other basic differentiation rules like this one that I will cover in my next posts.  Learn all of these rules well, and you'll have no problem differentiating complicated functions!

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