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## Main Question or Discussion Point

I had a second order differential equation where [itex]\psi[/itex] is the unknown function and it is a function of [itex]x[/itex]. We then introduced the following change of variable [itex]x = \sqrt{\frac{\hbar}{m \omega}} \xi[/itex]. When all was said and done I found that,

[tex] \frac{d^2 \psi}{d \xi^2} = \bigg(\frac{dx}{d \xi}\bigg)^2 \frac{d^2\psi}{dx^2} [/tex]

My question is, given an arbitrary change of variable for x and given an arbitrary order of the differential equation will the following formula always work?

[tex] \frac{d^n \psi}{d \xi^n} = \bigg(\frac{dx}{d\xi}\bigg)^n \frac{d^n \psi}{dx^n} [/tex]

[tex] \frac{d^2 \psi}{d \xi^2} = \bigg(\frac{dx}{d \xi}\bigg)^2 \frac{d^2\psi}{dx^2} [/tex]

My question is, given an arbitrary change of variable for x and given an arbitrary order of the differential equation will the following formula always work?

[tex] \frac{d^n \psi}{d \xi^n} = \bigg(\frac{dx}{d\xi}\bigg)^n \frac{d^n \psi}{dx^n} [/tex]