[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].