U1S4V05 Using the Delta definition.txt

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And we're back.
So we were developing this new notation for the derivative.
And we needed to figure out how to take this limit.
So hopefully you thought this through and deduced
that we want these two points to move closer and closer
together.
And so if that's the case, then we
should be taking the limit of this quotient as delta
x goes to zero.
So this is our complete alternative notation
for the derivative.
It's the limit of f of x plus delta x, minus f of x-- which
is the change in f-- over delta x-- the change in x-- as delta
x goes to zero.

Let's try it out.
So I'll erase this stuff.
And we'll look at the function f of x equals 1 over x.
So just the reciprocal function.
What's f prime of x?
Well, let's just plug this stuff in.
So we're going to get the limit as delta x approaches zero of--
and then our denominator is going to be delta x.
And for the numerator we need f of x plus delta
x-- so that's going to be the reciprocal, 1
over x plus delta x-- minus f of x, which is just 1 over x.
Now comes the algebra.
So we'll have the limit as delta x approaches zero.
Our denominator will still be delta x.
And our numerator, well, we've got these two fractions.
Let's just put them over a common denominator.
So our common denominator is going
to be x plus delta x times x.
And then when we do that, then the numerator is now going
to be x minus x plus delta x.

Let's keep going.
So we'll have the limit as delta x approaches zero.
Our denominator will still be delta x.
And now when we combine these terms up here,
we're going to have our x plus delta x times x.
And when we do the subtraction, we'll just get minus delta x.
And we're going to get some cancellation.
So that's good.
We're going to get the limit as delta x approaches zero.
And now we're just going to have a minus 1
over this x plus delta x times x.
And notice that our denominator is not going to zero anymore.
We've got as delta x approaches zero,
the denominator is just approaching x squared,
which is good.
And our numerator is of course just minus 1.
So we're going to get minus 1 over x squared as this limit.