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Now we can make the change of variables ##y\equiv\sqrt ax## to give ##f(y)=y^2##. Then ##d^2f/dy^2=2##.

It follows that

##\frac{d^2f}{dx^2}=a\frac{d^2f}{dy^2},##

but I can't replicate this with the chain rule.

I would put

##\frac{df}{dy}=\frac{df}{dx}\frac{dx}{dy}=\frac1{\sqrt a}\frac{df}{dx}##

##\frac{d^2f}{dy^2}=\frac{d^2f}{dx^2}\frac{dx}{dy}+\frac{df}{dx}\underbrace{\frac{d^2x}{dy^2}}_0=\frac1{\sqrt a}\frac{d^2f}{dx^2},##

which is a factor of ##1/\sqrt a## different from what we know the answer should be. So what am I doing wrong?

Thanks in advance!