2nd order differential substitution theory help please

Hello everyone.

So I don't seem to be understanding what is happening when I differentiate and as such I am getting into some serious muddles

So let's say we have a function of x, V, and y=V*x


we have a differential equation of y in terms of x ... all good so far, hopefully?

So what I do then is I then do dy/dx (so I can sub it into my equation)

dy/dx = V + dv/dx (by the product rule)

and this is where I can't seem to progress.

I know that d^2y/dx^2 = 2dv/dx + d^2v/dx^2 but I don't understand why that is true.


When I differentiate the dv/dx wrt x ... as far as I can see, dv/dx is an implicit function.

This is relevant to me because If I had g^2 ... and I differentiate it, I get 2g dg/dx ... (I know why that is true)

So I don't understand why the chain rule stops "working" or something? What am I missing?


Why is d/dx (dv/dx) not d^2v/dx^2 * dv/dx ?

Bold is your problem, you've misapplied the product rule. I'll just tell you because it should be obvious when you see it anyway: $\frac{dy}{dx} = V + x \frac{dV}{dx}$

I think you're missing an x from the thing you "know" as well.

Oh cheers. That was merely information I meant to add, sorry.

Still doesn't clear up the problem about how to differentiate the dv/dx term though ...