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## Exponential functions differentiation

# Derivatives of sin(x), cos(x), tan(x), eˣ & ln(x)

## Video transcript

Let's get some exposure
to the derivatives of some of the most
common functions. We're not going to prove
them in this video, but at least understand
what the derivatives are. So first, let's start
with the trig functions. If I want to take the derivative
with respect to x of sine of x, this is going to be
equal to cosine of x. And if you look at their graphs,
it'll make intuitive sense. Once again I have
not proved it here, but this is a good
thing to know, that the derivative of
sine of x is cosine of x. Now what about the
derivative of cosine of x? What about the derivative with
respect to x of cosine of x? Well, this one's going
to be negative sine of x. So the derivative
of sine is cosine, and the derivative
cosine is negative sine. And then finally, the
derivative of tangent of x is equal to 1
over cosine squared of x, which is equal to
the secant squared of x. Once again, these are all
very good things to know. Now let's talk a little
bit about exponentials and logarithms. So the derivative--
and actually, this is one of the coolest
results, and it once again speaks to how cool e is as
a number, the derivative with respect to
x of e to the x-- we need a drum
roll for this one. This is one of the coolest
things in mathematics. The derivative of e to
the x is e to the x. Now what does that tell us? And I have to take
a little pause here, because this is
just so exciting. So let me graph e to the x. So that's my y-axis. Let's say that this right
over here is my x-axis. So if I have very
negative values of x, e to a very negative value,
we are approaching zero. And then e to the
0 is 1, so that's going to be 1 right over there. So it's going to look
something like this. And then it's an exponential. It's going to go, it's going
to start increasing really, really, really,
really, really fast. So let's say that's the graph
of y is equal to e to the x. What this tells us is that
at any point-- so let's say I go right over here. I say when x is equal
to 0, e to the 0 is 1, what's the slope
of the tangent line here? Turns out that is also 1. Amazing. If I go to x is equal
to 1 right over here, the function evaluated here gets
us e to the 1 power or just e. And what's the slope of the
tangent line right over here? It is also e. At any point right over here,
the slope of the tangent line is equal to the value of
the function at that point. This is amazing. This is what is so cool about e. Anyway, that's not the
point of this video. This video is to give
you a catalog of all of the derivatives that
you might really need. So then finally,
if we're thinking about the derivative
with respect to x of the natural log of x,
this is going to be equal to-- and this is also fascinating. This is equal to 1 over
x or x to the negative 1. So somehow, we have
our natural log has kind of inserted
itself into-- when you take the derivative,
as filling in the gap that the power rule
left vacant, which is, is there some function
whose derivative is equal to x to the negative 1? The power rule gave us
functions whose derivatives might be x to the negative
2, x to the negative 3, or x to the squared
or x to the fifth. But it left the x to
the negative 1 vacant, and it's filled by
the natural log of x. Now I haven't proved it here. I've just catalogued
these for you. And then we can use
these in future videos, and we'll prove them
in future videos.