U2S2V03 What the derivative of a product is NOT.txt

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Let's start with these two functions.
They're pretty simple.
We know their derivatives, for instance.
Those derivatives are right here.
And we could add these two functions.
So if we did that, then we would get this right here,
f of x plus g of x.
And we could try to differentiate this sum.
So d by dx of f of x plus g of x,
well, we know how to do this.
We can just put 4x cubed here and add cosine x.
In other words, just take f prime plus g prime,
and that would be our answer.
So the derivative of the sum is the sum of the two derivatives.
Great.
What else might we do?
I suppose we could take h of x to be
equal to x to the fourth times sine
x, in other words, the product of f of x with g of x.
What would the derivative of this be?
It would be very tempting to simply say, well, let's just
put 4x cubed here and put cosine x there.
In other words, take f prime of x times g prime of x.
What's wrong with this?
Well, what's wrong with it is that h prime of x just
is not equal to f prime times g prime.
Now, this comes as a surprise to a lot of students.
And if you're surprised, that's understandable.
But it turns out that there's a fundamental reason why it's not
equal to f prime times g prime.
So instead of me just telling you what the right answer is,
we're first going to try to think through what
that fundamental reason is why this is not just going to be
equal to f prime times g prime.
And in the process of doing that,
we're actually going to discover some justification for what
the right answer actually is.
So that's the plan, and let's get started.