U2S3V08 Proof of the power rule for negative integer powers.txt

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I'm going to give a proof of the power rule
for negative integer powers using the quotient rule.
So to do this, I'm going to take the derivative of h
of x equal to x to the negative n
where n is a positive integer.
Because I want to use the quotient rule,
I'm going to rewrite this as a fraction,
so this is equal to 1 over x to the n and this is of the form
f of x over g of x where I'm letting f of x equal 1
and g of x equal x to the n.
Because I want to give a proof, I
can only use facts that I've already proven
and so that means that I can only
use the power rule on a function x
to the n for n, a positive integer,
and that's exactly what I have here with g.
So let's go ahead and find the derivative of h.
The quotient rule tells us that h prime
is equal to f prime times g minus g prime times f all
over g squared, which is equal to f prime, which
is 0 the derivative of 1, times g, which is x to the n,
minus the derivative of g and this
is where we're invoking the power
rule for positive integers.
This is n times x to the n minus 1 times
f, which is 1, all over x to the n quantity
squared, which is x to the 2n.
Simplifying this, I get that this
is equal to negative n times x to the n minus 1 minus 2n and n
minus 2n is negative n, so this is equal to negative n times
x to the negative n minus 1.
So let's just go ahead and rewrite that.
That means that we have just shown using the quotient rule
that the derivative with respect to x of x to the negative n
is equal to negative n times x to the negative n minus 1
and we've just said that this holds
for n, a positive integer, because that makes
this power a negative integer.
But let's go ahead and compare this
to the formula we had for the power
rule for positive integers.
This said that the derivative with respect to x of x to the n
is equal to n times x to the n minus 1.
This formula is exactly equal to the formula we just found if we
let n be a negative integer.
That means that this formula holds for n a positive integer
and a negative integer and in fact
it even holds when n is equal to 0, because x to the 0 is 1
and we know that the derivative of a constant is going to be 0.
So we've now proven that this formula holds for n equal to 0
plus or minus 1, plus or minus 2 for n, any integer.
We aren't done.

As we develop more rules and more techniques
for finding derivatives, we'll be
able to prove the power rule in more and more general forms
until we can actually give you the proof of the power rule
where n is any real number.