Derivative of a partial derivative

[Chicago_Boy1]

Hello,

So I have the function U(x,y). I have to find a partial derivative of U with respect to x. I understand that one can write that as U subscript x. But now I have to take d/dx of Ux, i.e. I have to take the derivative of Ux(x,y) with respect to x.

Supposedly the answer is Uxx + Uyx(dy/dx), but I don't really understand why...

Note that Ux = U subscript x
Note that Uyx = U subscript yx

Thank you in advance!

 

[HallsofIvy]

Here, in order to be able to talk about df/dx, we have to assume that f is a function of a single variable, x. But to talk about $\partial U/\partial x$ we must have U a function of two variable, x, and, say, y.

To be able to put them together, we must be thinking of y itself as a function of x.

Perhaps it would make more sense to separate the two uses of "x". Think of U as a function of x and y, and then think of x and y as functions of a parameter, t.

If f is any differentiable function of x and y and x and y themselves are differentiable functions of t, by the chain rule,
$$\frac{df}{dt}= \frac{\partial f}{\partial x}\frac{dx}{dt}+ \frac{\partial f}{\partial y}\frac{dy}{dx}= f_x\frac{dx}{dt}+ f_y\frac{dy}{dt}$$.

But here x= t so that dx/dt= 1 and dy/dt= dx/dt:
$$\frac{df}{dx}= \frac{\partial f}{\partial x}+ \frac{\partial f}{\partial y}\frac{dy}{dx}= f_x+ f_y\frac{dy}{dx}$$

Replace f by $U_x$ and you have your result.