U2S3V05 The Quotient Rule and Worked Example.txt

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Putting all the algebra together that you've just worked out,
you took the limit of delta h over delta t
as delta t went to 0 and what we found was that if h of t
is equal to f of t over g of t, then the derivative h prime
of t is given by this formula.
So it's the derivative of the top
times the bottom minus the top times the derivative
of the bottom all over the bottom function squared.
And this formula is going to hold wherever
f prime and g prime both exist and the denominator g of t
is not 0.
Remember this is only valid if everything in this formula
make sense and this formula is exactly what we're
going to call the quotient rule for derivatives.
Let's go ahead and use this in an example,
and one of my favorite examples is the tangent function.
Tangent of x is equal to sine x over cosine x.
The derivative with respect to x of the tangent function
is going to be equal to the derivative of the top function,
so the derivative of sine is cosine
times the bottom function, which is cosine again
minus the top function sine of x times the derivative
of the bottom function and the derivative of cosine
is negative sine-- don't forget that negative-- and all
over the denominator squared, so cosine squared on the bottom.
This numerator is cosine squared plus sine squared, which
is equal to 1, so we can simplify this to 1
over cosine squared and I hope you remember this
if not you should probably review
your trig functions a little bit, but another name for this
would be the secant squared of x because the secant is defined
as 1 over cosine.
This is pretty exciting and there are a whole lot
more examples of quotient functions
that now you can solve, so I want
you to go ahead and get some practice and then we'll be back
and we'll work through some more examples together.