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## AP®︎ Calculus BC (2017 edition)

### Unit 4: Lesson 7

Derivatives of logarithmic functions- Derivative of ln(x)
- Derivative of logₐx (for any positive base a≠1)
- Worked example: Derivative of log₄(x²+x) using the chain rule
- Differentiate logarithmic functions
- Differentiating logarithmic functions review
- Common derivatives review
- Proof: the derivative of ln(x) is 1/x
- Worked example: Derivative of ln(√x) using the chain rule
- Derivative of sin(ln(x²))
- Differentiating logarithmic functions using log properties

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# Derivative of ln(x)

AP.CALC:

FUN‑3 (EU)

, FUN‑3.A (LO)

, FUN‑3.A.4 (EK)

The derivative of ln(x) is 1/x. We show why it is so in a different video, but you can get some intuition here.

## Video transcript

- [Instructor] In this
video, we're going to think about what the derivative
with respect to x of the natural log of x's. And I'm gonna go straight
to the punch line. It is equal to one over x. In a future video, I'm
actually going to prove this. It's a little bit involved. But in this one, we're
just going to appreciate that this seems like it is actually true. So right here is the graph of y is equal to the natural log of x. And just to feel good about the statement, let's try to approximate what the slope of the tangent
line is at different points. So let's say right over here, when x is equal to one, what does the slope of the
tangent line look like? Well, it looks like here, the slope looks like it is equal, pretty close to being equal to one, which is consistent with the statement. If x is equal to one, one
over one is still one, and that seems like what
we see right over there. What about when x is equal to two? Well, this point right over
here is the natural log of two, but more interestingly,
what's the slope here? Well, it looks like, let's see, if I try to
draw a tangent line, the slop of the tangent line
looks pretty close to 1/2. Well, once again, that is one over x. One over two is 1/2. Let's keep doing this. If I go right over here,
when x is equal to four, this point is four comma
natural log of four, but the slope of the tangent line here looks pretty close to 1/4 and if you accept this, it is exactly 1/4, and you could even go
to values less than one. Right over here, when x is equal to 1/2, one over 1/2, the slope should be two. And it does indeed, let me do this in a slightly different color, it does indeed look like
the slope is two over there. So once again, you take the derivative with respect to x of the natural
log of x, it is one over x. And hopefully, you get a sense that that is actually true here. In a future video,
we'll actually prove it.