U2S3V04 Approach to finding the quotient rule.txt

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So we're considering an example where
we're letting the function h of t
be a quotient of two functions.
So quotient f of t divided by g of t.
And we're interested in finding the derivative h prime of t.
It's going to take a little while to figure out
what this is, but we've already figured out what it can't be.
We know that h prime of t is not equal to f prime over g prime,
because the units do not agree.
So we're now going to outline the approach
for finding the derivative of the quotient, which
is going to be the same approach that we used in the product
rule, but this time, we're going to ask you to carry
these steps out on your own.
The approach is to use a linear approximation.
So what I'm going to do is I'm going
to say I know that h prime of t is very close to being
the same thing as delta h over delta t when delta t is small.
So that's the first step.
The second step is to go ahead and identify delta h in terms
of f, delta f, g, and delta g.
And the third and final step is to take
the limit of this expression that we
get in terms of just f, g, and t as delta t approaches 0.
And this is going to give us a formula for the derivative
of a quotient.

This procedure that we worked through with the product rule,
and we've just outlined here is extremely general,
and that's why we want you to work
through these steps on your own exactly because this
is a valuable problem solving technique.
In fact, we are going to use it repeatedly in this unit.
And it's used all over science, mathematics, and engineering
in both continuum modeling and discrete numerical modeling
no matter how complicated the system.

So why don't you get started.
And we'll help out with some of the trickier algebraic
manipulations.
And we'll see you on the other side.