I stumbled across this result:
$$f_{xx}(ax+by) = \frac{a}{b} f_{xy}(ax+by)$$
Which I can't off the top of my head justify... I'm sure it's a very simple property but I can't seem to be able to convince myself of it. How would one go about showing this? Assume $f$ is $C^2$.
Assuming $f$ is twice differentiable one has $$f_{x}(ax+by)=f'(ax+by)\cdot a$$ Differentiating one more time wrt $x$ yields $$f_{xx}(ax+by)=f''(ax+by)\cdot a^2$$ Now we differentiate with respect to $y$ to get $$f_{xy}(ax+by)=f''(ax+by)\cdot ab$$ Therefore $$\frac{1}{a^2}f_{xx}(ax+by)=\frac{1}{ab}f_{xy}(ax+by)$$ The result follows.