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## AP®︎/College Calculus BC

### Unit 2: Lesson 8

Derivatives of cos(x), sin(x), 𝑒ˣ, and ln(x)- Derivatives of sin(x) and cos(x)
- Worked example: Derivatives of sin(x) and cos(x)
- Derivatives of sin(x) and cos(x)
- Proving the derivatives of sin(x) and cos(x)
- Derivative of 𝑒ˣ
- Derivative of ln(x)
- Derivatives of 𝑒ˣ and ln(x)
- Proof: The derivative of 𝑒ˣ is 𝑒ˣ
- Proof: the derivative of ln(x) is 1/x

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# Derivative of 𝑒ˣ

AP.CALC:

FUN‑3 (EU)

, FUN‑3.A (LO)

, FUN‑3.A.4 (EK)

The derivative of 𝑒ˣ is... well... 𝑒ˣ. This is a very special property lies at the heart of our work with exponential functions.

## Video transcript

- [Instructor] What we
have right over here is the graph of Y is equal to E to the X and what we're going to know
by the end of this video is one of the most
fascinating ideas in calculus and once again it reinforces the idea that E is really this
somewhat magical number. So we're gonna do a little
bit of an exploration. Let's just pick some points on this curve of Y is equal to E to the X and think about what the
slope of the tangent line is or what the derivative looks like and so let's say when Y is equal to one or when E to the X is equal to one, this is the case when X is equal to zero. Well, the slope of the tangent line looks like it is one,
which is curious because that's exactly the value of
the function at that point. What about when E to the X is
equal to two right over here? Well here, let me do it in another color, the slope of the tangent
line sure looks pretty close, sure looks pretty close to two. What about when E to
the X is equal to 1/2? So that's happening right about here. Well, it sure looks like the
slope of the tangent line is about 1/2. We could try what happens when E to the X is equal to five? Well, the slope of the tangent line here sure does look pretty close, sure does look pretty close to five and so just eyeballing it, is it the case that the
slope of the tangent line of E to the X is the same
thing, is E to the X? And I will tell you and
this is an amazing thing that that is indeed true, that if I have some function, F of X, that is equal to E to the X and if I were to take
the derivative of this, this is going to be equal
to E to the X as well or another way of saying it, the derivative with
respect to X of E to the X is equal to E to the X and that is an amazing thing. In previous lessons or courses, you've learned about ways to define E and this could be a new one. E is the number that where
if you take that number to the power of X, if
you define a function or expression as E to the X, it's that number where if you
take the derivative of that it's still going to be E to the X. And what you're looking here, this curve, it's a curve where the
value that's Y value at any point is the same as
the slope of the tangent line. If that doesn't strike you as mysterious and magical and amazing just yet, it will. Maybe tonight you'll wake up
in the middle of the night and you'll realize just what's going on. Now, some of you might be
saying okay, this is cool, you're telling me this, but
how do I know it's true? In another video, we will do the proof.