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