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Current time:0:00Total duration:2:03

AP.CALC:

FUN‑3 (EU)

, FUN‑3.A (LO)

, FUN‑3.A.4 (EK)

so in this video we're going to think about what the derivative with respect to X of the natural log of X is and I'm going to go straight to the punch line it is equal to 1 over X and 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 take the slope 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 1 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 1 which is consistent with this statement if X is equal to 1 1 over 1 is still 1 and that seems like what we see right over there what about what x is equal to 2 well this point right over here is the natural log of 2 but more interestingly what's the slope here well it looks like let's see if I try to draw a tangent line the slope of the tangent line looks pretty close to 1/2 well once again that is 1 over X 1 over 2 is 1/2 let's keep doing this if I go right over here when X is equal to when X is equal to 4 this point is 4 comma natural log of 4 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 1 right over here when X is equal to 1/2 1 over 1/2 the slope should be 2 and it does indeed let me do it in a slightly different color it does indeed look like the slope is 2 over there so once again you take the natural log you take the derivative with respect to X of the natural log of X it is 1 over X and hopefully you get a sense that that is actually true here in a future video will actually prove it

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