Differentiation Rules - Finding the Derivative of a Sum of Functions

Welcome back to my introductory calculus series on differentiation formulas.  For those who are playing along at home, I have explained several rules so far and am going to add another one today.  If you've missed those posts, then I highly encourage you to go back and take a look at them to familiarize yourself with these basic concepts.  (So far: here, here, and here.  Or just check my table of contents to find more!)  Today, I am going to talk about finding the derivative of a sum of functions.

Consider that you have two functions.  They can be whatever you want, it really doesn't matter.  While I explain this concept, let's just call these functions f(x) and g(x).  Now, suppose that you want to add these functions together, and you come up with a third function, let's call it F(x).  Sounds complicated?  It doesn't have to be.  It's actually quite simple.  But what about if we want to find the derivative F'(x) - how do we do that??  That's easy too.  You just need to understand this property of derivatives.

All you need to do is find the derivatives of the "smaller" functions, f'(x) and g'(x), and then add those together to get F'(x)!

Put simply, the derivative of a sum of functions is equal to the sum of those functions' derivatives.

In other words: F'(x) = f'(x) + g'(x).

Let's try an example, and put some numbers to this thing so that you can see it's not that crazy.

Keep in mind that when I talk about functions here, these could be anything: x², or (x-2)^5/16. Simple, or more complicated.  The same rules apply.  For now, let's keep it simple.

Suppose that f(x) = x² and g(x) = x³.  What is the sum of these, F(x)?

Well this is as easy as it gets.  What is the sum of x² + x³?
Your new function F(x) is just that: F(x) =x²+ x³.

So then, differentiate F(x) to find its derivative, F'(x).

By the rule that I explained above, all you have to do is find the derivative of the individual pieces of this sum, and then add those!  So, by the power rule, we have:

F'(x) = 2x + 3x²

That's it!

Obviously this can get a heck of a lot more complicated, but the principle remains the same. What if you want to find the derivative of some G(x) = x⁴ + x² + 4x^2/3.  Think of this as a sum of three smaller functions, and you can see how this rule applies here.  You don't need to necessarily always have two distinctly different functions to apply this.  Think back to what a function is - they can have many terms to them.  (They just have to pass the vertical line test!)

My point in all of this rambling is that to find derivatives of functions that have multiple terms in them added together (like G(x) above, or F(x) before that), all you need to do is find the derivatives of the individual terms, and add them together.  It really is quite straightforward, and I hope that I have made helped to make this clear.  Once you get in the habit of applying this rule, you will do it automatically.

Please let me know if this has helped you, or should be clearer in some way.  I appreciate any feedback you would like to leave!  Also, please do me a favour and click on the Like or +1 buttons if this post helped you in any way.

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