One Rule To Add Them All: Derivations of Sums

So, we know how derivatives work, which is fantastic.  So far, we've only looked at one very simple example.  From our previous work, we found that when   it means 
We'll take a quick look at another simple example, then we'll add them together, then work out a proof that works for all sums.

Our second example will be .  So, we know that .  Further, we know that this means that .  Plugging in our value for which is amusingly just x, we can say with

certainty that 

Simplifying, we are left with .  At this point, it's safe to say that no longer has any influence on our function so the limit as approaches is simply .

We can now say that when it means


So, what happens when we combine the two?  Let's say our new function is

We know that  

A little bit of expansion and this becomes

Now we simplify and come up with

Then we clean up our denominator yielding:

Solve for our limit by plugging in 0, and we are left with


The astute among you will notice that as our original function was , the resulting derivative is equal to


We can prove that this is true for all sums with relative ease:

Given that   we can see simply that the change in or is equal to

In terms we are familiar with , this means

Next we divide both sides by h giving us 

Finally, we apply the limit as h approaches 0 to both sides

That form should look familiar. In fact we can see that

and

and

Which formalizes as 



We now know that anytime we are provided with a function with sums (it works with subtraction too, just think about it) we can derive each piece of the sum individually.