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[MUSIC]. 
Looks like I've got two functions f(x) 

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and g(x), and they're both differentiable 
at a. 

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Then I can define a new function. 
h(x) which is the sum of f and g. 

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Alright it's a new function, to compute 
h(x) I just plug x into f and I plug x 

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into g and I add together whatever f and 
g give me. 

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Alright so that's a new function that I 
build from f and g. 

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Now here's the conclusion, right? Then 
each prime of a is just the sum of the 

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derivative of f at a and the derivative 
of g at a. 

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And to prove something like this, this is 
a really a theorem, right? This is a 

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theorem that tells me how to compute. 
The derivative of the sum of functions. 

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And how do I prove something like this? 
Why I just go back to the definition of 

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derivative. 
Alright. 

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The derivative of this function h at the 
point a is the limit as x goes to a of, 

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h(x)-h(a)/x-a. 
Now I know what h(x) is. 

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h(x) is f(x) + g(x). 
So I can plug that in. 

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Alright, so this is the limit as x goes 
to a of f(x)+g(x). 

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And I also know what h(a) is, right? I 
just plug in a for x. 

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And I get that h(a) is f(a)+g(a). 
And this is all divided by the same 

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denominator, x-a. 
Great. 

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I want to calculate that limit, right?. 
Well, I can rearrange the numerator, so 

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the numerator is the same as what? This 
is f(x) + g(x) - f(a). 

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Minus g of a, but rearrange the numerator 
and get f of x minus f of a plus g of a x 

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minus g of a and this is divided by x 
minus a. 

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Now what do I do? Well I can actually 
split this up into 2 separate fractions, 

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alright? This is f(x)-f(a)/x-a, 
g(x)-g(a)/x-a. 

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So this is the limit of 
f(x)-f(a)/x-a/x-a+g(x)-g(a)/x-a). 

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That's a As a limit is x goes to a How do 
I calculate that limit. 

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Okay. 
I'm just applying these, these rules for 

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calculating limits and one for the rules 
of calculating limits is the limit of the 

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sum is the sum of the limits provided the 
limits exist. 

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What are these 2 limits? Well, this is 
really the derivative of f(a) and this is 

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really the derivative of g(a). 
And I assume that f and g are both 

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differentiable at a. 
So, those limits do, do exist and I can 

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apply the limit of the sum and the sum of 
the limits. 

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So this = the limit as x goes to a of 
f(x)-f(a)/(x-a) + the limit as x goes to 

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a of g(x)-g(a)/x-a, because I know those 
2 limits exist. 

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And I even know what they're equal to, 
right? I have a name for those 2 limits. 

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This 1st limit is the derivative of f at 
a, this 2nd limit is the derivative of g 

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at a. 
So this is f prime of a plus g prime of a 

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and that's exactly what I wanted to 
show,right? I wrote down the definition 

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of derivative of h at the point a, there 
is is and I applied properties of limits 

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until I conclude that that limit is 
equal. 

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To the derivative of f(a) + the 
derivative of g(a), alright? And this is 

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what tells me how to calculate the 
derivative of a sum. 

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Alright, if I've got a sum of 2 
functions, this is telling me that as 

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long as those 2 functions are both 
differentiable at a, I can calculate the 

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derivative by just adding together the 
derivatives of f and g. 

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And hopefully this, this should seem 
reasonable, right? Because what is the 

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derivative measuring, right, it's 
measuring how much change in the input 

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changes the output. 
Right, I want to know how much wiggling 

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the input a, would effect the output of 
H, and that's what this derivative is 

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measuring. 
Right? Well, that's really going to be, 

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you know, somehow connected to how 
wiggling the input to f changes f and 

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wiggling the input to g changes g and I'm 
just adding them together. 

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So I think this makes sense that, then 
the, how the output changes which would 

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be the sum of how these 2 component 
functions change. 

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[MUSIC].