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!

One final comment - if you thought this post was helpful in any way, then please do me a favour and click on the Like and +1 buttons found on this page!  Thanks for your support and for visiting.  Be sure to come back if you need help with any other maths concepts.

Top 5 Most Popular Posts of April

It's that time of month again - time for a recap of my top 5 most popular posts of April!  Once again, the top 5 are dominated by several of the usual favourites.  However, spot number 5 is a newcomer!  I'm happy to see that I have several pieces of content that are so routinely visited, but I also am very pleased to see new posts crack the top 5 as well from time to time.  If you missed these stories when they were originally posted, now is a great time to catch up on what so many people think are some of the best!

Before you take a look at these, please allow me to invite you to check out my recent series on Differentiation.  You can find an introduction to derivatives and differential calculus here, and then follow-up posts on an elementary differentiation technique and the power rule.
1. Stretching and Compressing Graphs.  Long live the king!  This is far and away one of my most popular posts, ever since its original posting nearly 6 years ago!  Pay it a visit to see what everyone is talking about.

2. Converting Point-Slope Form to Standard Form.  Another popular post, explaining how to go about converting your equation from point-slope to into standard form.  Along with slope-intercept form, these are the most common ways of expressing equations easily so that they can be graphed, so it's essential to understand how to convert between them.

3. Which Measure of Central Tendency to Use? Mode, Mean, or Median?  Do you know the differences between these three measures of "center?"  Where would you use a mean, and when would it be most appropriate to use median or mode.  Read my post to get some tips to help you choose the best method!

4. Special Angles in Trigonometry.  If you memorize - or better yet, understand - what these special angles are and how you can find them, then they will make working with triangles in trigonometry a lot easier!

5. Graphing - Parallel and Perpendicular Lines.  Graphing parallel or perpendicular lines is a very common task in learning mathematics.  In this post, I explain just what these terms mean, and specifically answer the question of how to determine if two lines are exactly perpendicular or not.
To find more great explanations and discussions of math concepts on my site, browse the Math Concepts Explained table of contents.  Alternately, you can enter your topic of interest in the search bar at the top of every page.

If you enjoy Math Concepts Explained, I invite you to join the many other students, teachers, and math enthusiasts who follow my site:
Thanks to all of my visitors for your support!