How can i calculate $d/dt(\partial f/\partial x)$

Question

Let $f$ be a function of two variables such that $$x=a+th \text{ and } y=b+tk$$

Then $$\frac{d}{dt}\left(\frac{\partial f}{\partial x}\right)= \frac{dx}{dt}\frac{\partial^2 f}{\partial x^2} + \frac{dy}{dt}\frac{\partial^2 f}{\partial x\partial y}$$

But where did this rule come from? It’s obvious that it is related to the chain rule but how exactly ?

Answer 1

hint

What you might know is :

If $ F $ is a function of two variables $ x $ and $ y $, then $$\boxed{dF=\frac{\partial F}{\partial x}dx+\frac{\partial F}{\partial y}dy}$$

You just need to replace $ F(x,y) $ by ${\partial f}/{\partial x }$ and divide by $ dt $.

The equation you wrote is not correct. The last term should be $$\frac{dy}{dt}\frac{\partial^2 f}{\partial y\partial x}$$