NARRATOR: Exercise three.
Give an alternative proof
that the derivative of log e of x is one divided by x
by differentiating both sides of the equation
X equals e to the power of log e of x.
So we start by differentiating both sides of this equation.
The derivative of x is 1.
And the derivative of e to the power of log e of x
is going to require the use of the chain rule.
And the result of the chain rule is e to the power of log e of x
multiplied by the derivative of log e of x.
So as an aside,
the chain rule involves us
allowing y to equal
e to the power of log e of x.
So ultimately, we're trying to
calculate the derivative of this,
that is dy/dx.
And we'll then let
u equal log e of x.
And this then implies that
y is equal to e to the power of u.
And so the derivative dy/du
is also e to the power of u.
The chain rule states that
dy/dx is equal to
dy/du multiplied by du/dx.
And we've calculated dy/du
to be e to the power of u.
So substituting this in,
we also know that u is log e of x.
So e to the power of u is in fact
e to the power of log e of x
and du/dx will be the derivative
of log e of x,
which is what we're trying to prove.
So we'll leave this written
as the derivative of log e of x.
And so we get the equation that
1 is equal to
e to the power of log e of x
multiplied by the derivative of log e of x.
So e to the power of log e of x is equal to x.
And then simply dividing both sides of the equation by x
we get the final result that the derivative of log e of x
is in fact 1 divided by x.