[MUSIC]. 
Looks like I've got two functions f(x) 
and g(x), and they're both differentiable 
at a. 
Then I can define a new function. 
h(x) which is the sum of f and g. 
Alright it's a new function, to compute 
h(x) I just plug x into f and I plug x 
into g and I add together whatever f and 
g give me. 
Alright so that's a new function that I 
build from f and g. 
Now here's the conclusion, right? Then 
each prime of a is just the sum of the 
derivative of f at a and the derivative 
of g at a. 
And to prove something like this, this is 
a really a theorem, right? This is a 
theorem that tells me how to compute. 
The derivative of the sum of functions. 
And how do I prove something like this? 
Why I just go back to the definition of 
derivative. 
Alright. 
The derivative of this function h at the 
point a is the limit as x goes to a of, 
h(x)-h(a)/x-a. 
Now I know what h(x) is. 
h(x) is f(x) + g(x). 
So I can plug that in. 
Alright, so this is the limit as x goes 
to a of f(x)+g(x). 
And I also know what h(a) is, right? I 
just plug in a for x. 
And I get that h(a) is f(a)+g(a). 
And this is all divided by the same 
denominator, x-a. 
Great. 
I want to calculate that limit, right?. 
Well, I can rearrange the numerator, so 
the numerator is the same as what? This 
is f(x) + g(x) - f(a). 
Minus g of a, but rearrange the numerator 
and get f of x minus f of a plus g of a x 
minus g of a and this is divided by x 
minus a. 
Now what do I do? Well I can actually 
split this up into 2 separate fractions, 
alright? This is f(x)-f(a)/x-a, 
g(x)-g(a)/x-a. 
So this is the limit of 
f(x)-f(a)/x-a/x-a+g(x)-g(a)/x-a). 
That's a As a limit is x goes to a How do 
I calculate that limit. 
Okay. 
I'm just applying these, these rules for 
calculating limits and one for the rules 
of calculating limits is the limit of the 
sum is the sum of the limits provided the 
limits exist. 
What are these 2 limits? Well, this is 
really the derivative of f(a) and this is 
really the derivative of g(a). 
And I assume that f and g are both 
differentiable at a. 
So, those limits do, do exist and I can 
apply the limit of the sum and the sum of 
the limits. 
So this = the limit as x goes to a of 
f(x)-f(a)/(x-a) + the limit as x goes to 
a of g(x)-g(a)/x-a, because I know those 
2 limits exist. 
And I even know what they're equal to, 
right? I have a name for those 2 limits. 
This 1st limit is the derivative of f at 
a, this 2nd limit is the derivative of g 
at a. 
So this is f prime of a plus g prime of a 
and that's exactly what I wanted to 
show,right? I wrote down the definition 
of derivative of h at the point a, there 
is is and I applied properties of limits 
until I conclude that that limit is 
equal. 
To the derivative of f(a) + the 
derivative of g(a), alright? And this is 
what tells me how to calculate the 
derivative of a sum. 
Alright, if I've got a sum of 2 
functions, this is telling me that as 
long as those 2 functions are both 
differentiable at a, I can calculate the 
derivative by just adding together the 
derivatives of f and g. 
And hopefully this, this should seem 
reasonable, right? Because what is the 
derivative measuring, right, it's 
measuring how much change in the input 
changes the output. 
Right, I want to know how much wiggling 
the input a, would effect the output of 
H, and that's what this derivative is 
measuring. 
Right? Well, that's really going to be, 
you know, somehow connected to how 
wiggling the input to f changes f and 
wiggling the input to g changes g and I'm 
just adding them together. 
So I think this makes sense that, then 
the, how the output changes which would 
be the sum of how these 2 component 
functions change. 
[MUSIC].