Welcome to a proof of the derivative
of f of x equals e to the x.
We'll be using the limit
definition of the derivative
to prove the derivative of e
to the x, with respect to x,
equals e to the x.
What's really quite amazing,
this function is its own derivative.
So to begin,
we'll apply the limit
definition of the derivative.
So we have the derivative of
e to the x, with respect to x,
equals the limit as h approaches zero of,
for f of the quantity of x plus h,
we would have e raised
to the power of x plus h.
Then for minus f of x,
we have minus e to the x
and all this is divided by h.
Now for the next step,
we're going to use a property of exponents
to factor the numerator.
Remember when we're multiplying,
the bases are the same,
we add the exponents,
and therefore e raised
to the power of x plus h,
is equal to e to the x times e to the h,
and therefore we have a common factor
of e to the x in the numerator.
So if we factor the numerator,
we have the limit as h
approaches zero of e to the x
times the quantity e to the h minus one.
Let's just check this by distributing
e to the x times e to the h
is equal to e to the power of x plus h.
And then we have minus
e to the x times one,
which would give us e to the x.
Now because we have a
product in the numerator,
we'll write this as a limit of a product.
So now we have the limit
as h approaches zero
of e to the x times
the quantity e to the h
minus one, divided by h.
Now because we have a limit of a product,
we can write this as a product of limits.
So we have the limit as h approaches zero
of e to the x times the
limit as h approaches zero
of e to the h minus one, divided by h.
In this form, we can evaluate both limits.
The limit as h approaches
zero of e to the x
is just equal to e to the x,
because e to the x is not affected
as h approaches zero.
And then for the second limit,
we have the limit as h approaches zero
of e to the h minus one divided by h,
which we proved earlier is equal to one
using the Squeeze Theorem.
So we have e to the x times one,
which equals e to the x
and therefore we have our proof.
The derivative of e to
the x, with respect to x
equals e to the x.
Before we go though,
let's look at this graphically.
F of x equals e to the x
is graphed here in blue.
And we have the tangent line graphed
at the point one comma e.
Because f of x and f-prime of x are both
equal to e to the x,
we know the slope of this tangent line
is equal to the y coordinate
function value of e.
E is the function value
and the derivative function
value at x equals one.
So if we were to select any point
on f of x is equal to e to the x,
the function value will give us the slope
of the tangent line at that point.
For example, if we selected
this point on the function,
where we can see that
the y coordinate function
value is six,
the slope of the tangent
line at this point
would be six.
If we selected this point on the function,
where the y coordinate
function value is one,
the slope of the tangent
line at that point
would be one.
And again this is because
f of x equals e to the x
is its own derivative.
I hope you found this helpful.
