U1S4V09 Derivative of a Sum.txt

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In this video, we're going to talk about the derivative
of a sum of functions.
So suppose that h of x equals f of x plus g of x for all x.
It turns out that the derivative, h prime of x,
is going to equal f prime of x plus g prime of x, as long as f
prime and g prime both exist.
So it's very convenient.
The derivative of the sum is the sum of the two derivatives.
The same thing is true of differences.
If h is the difference f of x minus g
of x, then h prime of x is going to be f prime of x minus g
prime of x.
We're going to prove this in a separate video,
but for now let's just do an example.
Let's say that h of x equals 1 over x plus 3x minus 7.
So here we have h of x being the sum of two functions.
It's the reciprocal function 1 over x
added to this linear function 3x minus 7.
And so h prime of x is just going
to be the sum of the derivatives of those two pieces.
The derivative of 1 over x we've already done.
That's minus 1 over x squared.
And then we just add that to the derivative of this linear part,
the 3x minus 7.
And we know that the derivative of mx plus b is just m.
It's just the slope of that line.
So the derivative of 3x minus 7 is 3, and that's our h prime.
We've just taken the derivative of this part
and added it to the derivative of this part.
Nice and simple.
So why don't you try some on your own?