Suppose I have a function $f$ and want to differentiate it with respect to $x^2$ at the point $x=x_0$. Is the below true:
$$\frac{d f}{d x^2}\bigg|_{x=x_0}=\bigg(\frac{d x}{d x^2}\frac{d f}{d x}\bigg)\bigg|_{x=x_0}$$ $$=\bigg(\frac{1}{2x}\frac{d f}{d x}\bigg)\bigg|_{x=x_0}$$ $$=\frac{1}{2x_0}\bigg(\frac{d f}{d x}\bigg)\bigg|_{x=x_0}$$
If so/if not, why/why not?
Yes it is. Let $y=x^2.$ Then $$\frac{dy}{dx}=2x.$$ Now$$\frac{df}{dx^2}=\frac{df}{dy}=\frac{df}{dx}\frac{dx}{dy}=\frac{1}{2x}\frac{df}{dx}.$$