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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

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 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 want to take the derivative of that so we then want to take the derivative of that to get us our second derivative and so 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 and to do that let's just remind ourselves that we just have two 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 6 X to the negative 2 so let's take the derivative of both sides of this with respect to X so 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 is X to the negative 3 and actually let me give myself a little bit more space here so this is negative 12 X to the negative 3 and now let's take the derivative of that with respect to X so I'm going to 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 X is going to be equal to well we just have 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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