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Second derivatives | Advanced derivatives | AP Calculus AB | Khan Academy

2m read
·Nov 11, 2024

Let's say that Y is equal to 6 over x squared. What I want to do in this video is figure out what is the second derivative of Y with respect to X.

If you're wondering where this notation comes from for a second derivative, imagine if you started with your Y and you first take a derivative. We've seen this notation before, so that would be the first derivative. Then we want to take the derivative of that, so we then want to take the derivative of that to get us our second derivative. That's where that notation looks comes from. It looks like we're having you have a d squared d times d, although you're not really multiplying them.

Applying the derivative operator twice, it looks like you have a dx squared. Once again, you're not multiplying them; you're just applying the operator twice. But that's where that notation actually comes from.

Well, let's first take the first derivative of Y with respect to X. To do that, let's just remind ourselves that we just have to apply the power rule here. We can just remind ourselves, based on the fact that Y is equal to 6 X to the negative 2.

So let's take the derivative of both sides of this with respect to X. With respect to X, I'm going to do that, and so on the left-hand side, I'm going to have dy/dx is equal to, now on the right-hand side, take our negative 2, multiply it times the 6; it's going to get negative 12 X to the negative 2 minus 1, which is X to the negative 3.

Actually, let me give myself a little bit more space here. So this is negative 12 X to the negative 3. Now, let's take the derivative of that with respect to X. So I'm going to apply the derivative operator again.

The derivative with respect to X, now the left-hand side gets the second derivative of Y with respect to X is going to be equal to, well, we just used the power rule again. Negative 3 times negative 12 is positive 36 X times X to the, well, negative 3 minus 1 is negative 4 power, which we could also write as 36 over X to the fourth power.

And we're done.

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