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Second derivatives

AP.CALC:
FUN‑3 (EU)
,
FUN‑3.F (LO)
,
FUN‑3.F.1 (EK)
,
FUN‑3.F.2 (EK)

Video transcript

- [Voiceover] Let's say that y is equal to six over x-squared. What I wanna do in this video is figure out what is the second derivative of y with respect to x. And 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, and we've seen this notation before. So that would be the first derivative. Then we wanna take the derivative of that. So we then wanna take the derivative of that to get us our second derivative. And so that's where that notation comes from. It likes you have a d-squared, d times d, although you're not really multiplying them. You're applying the derivative operator twice. It looks like you have a dx-squared. Once again, you're not multiplying 'em, 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. And to do that, let's just remind ourselves that we just have to apply the power rule here, and we can just remind ourselves, based on the fact that y is equal to six x to the negative two. So let's take the derivative of both sides of this with respect to x, so with respect to x, gonna do that, and so on the left-hand side, I'm gonna have dy dx is equal to, now on the right-hand side, take our negative two, multiply it times the six, it's gonna get negative 12 x to the negative two minus one is x to the negative three. Actually, let me give myself a little bit more space here. So this negative 12 x to the negative three. And now, let's take the derivative of that with respect to x. So I'm gonna apply the derivative operator again, so the derivative with respect to x. Now the left-hand side gets the second derivative of y with respect to to x, is going to be equal to, well, we just use the power rule again, negative three times negative 12 is positive 36, times x to the, well, negative three minus one is negative four power, which we could also write as 36 over x to the fourth power. And we're done.