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# Derivative of 2ˣ (old)

An older video where Sal finds the derivative of 2ˣ using the derivative of eˣ and the chain rule. Created by Sal Khan.

Video transcript

Let's see if we can take
the derivative with respect to x of 2 to the x power. And you might say,
hold on a second. We know how to take the
derivative of e to the x. But what about a base like 2? We don't know what to do with 2. And the key here is
to rewrite 2 to the x so that we essentially
have it as e to some power. And the key there
is to rewrite 2. So how can we rewrite 2
so it is e to some power? Well, let's think about what e
to the natural log of 2 power is. The natural log
of 2 is the power that I would have to
raise e to to get to 2. So if we actually
raise e to that power, we are going to get to 2. So what we could do, instead
of writing 2 to the x, we could rewrite this as e. We could rewrite 2 as e
to the natural log of 2, and then raise that
to the x power. So this is the x
power in yellow. And so let's do that
right over here. So instead of taking the
derivative with respect to x of 2 to the x,
let's say, let's just take the derivative
with respect to x of the exact same
expression rewritten, of e to the natural log of
2 raised to the x power. Let me put this x in
that same color, dx. Now we know from our
exponent properties if we raise something
to some power, and then raise that
to another power, we can take the product
of the two powers. Let me rewrite this
just to remember. If I have a to b, and then
I raise that to the c power, this is the exact same thing
as a to the b times c power. So we can utilize that
exponent property right here to rewrite this as being equal
to the derivative with respect to x of e to the natural
log of 2 times x. And what's neat
about this is now we've got this into a form
of e to the something. So we can essentially use the
chain rule to evaluate this. So this derivative is going to
be equal to the derivative of e to the something with
respect to that something. Well, the derivative e to
the something with respect to that something is
just e to that something. So it's going to be equal
to e to the natural log of 2 times x. So let me make it clear
what I just did here. This right over here
is the derivative of e to the natural
log of 2 times x with respect to the
natural log of 2-- let me make it a little
bit clearer-- with respect to the natural log of 2 times x. So we took the derivative
of e to the something with respect to that something--
that's this right here, it's just e to that something. And then we're going
to multiply that by, this is just an application
of the chain rule, of the derivative of that
something with respect to x. So the derivative of natural log
of 2 times x with respect to x is just going to be
natural log of 2. This is just going to
be natural log of 2. The derivative of a times x is
just going to be equal to a. This is just the
coefficient on the x. And just to be clear,
this is the derivative of natural log of 2 times
x with respect to x. So we're essentially done. But we can simplify
this even further. This thing right over
here can be rewritten. And let me draw a
line here just to make it clear that this equals sign
is a continuation from what we did up there. But this e to the natural log
of 2x, we can rewrite that, using this exact same
exponent property, as e to the natural
log of 2, and then all of that raised
to the x power. And of course, we're multiplying
it times the natural log of 2, so times the natural log of 2. Well, what is e to
the natural log of 2? Well, we already
figured that out. That is exactly equal to 2. This right over
here is equal to 2. And so now we can simplify. This whole thing, the
derivative of 2 to the x, is equal to-- and I'll switch
the order a little bit-- it is the natural log of
2, that's this part right over here, times 2 to the x. Or we could write it as 2 to the
x times the natural log of 2.