# Interchanging total derivative and partial derivative

Say I have a function $F(x,y)$, where $x = f(t)$ and $y = g(t)$.

$$\frac{\mathrm{d} }{\mathrm{d} t} \frac{\partial F}{\partial x} \tag{1}$$

$$\frac{\partial }{\partial x} \frac{\mathrm{d} F}{\mathrm{d} t} \tag{2}$$

They both evaluate to the same thing. But is there something that I have to watch out for? For example, note that in eqn. (2), $\frac{\mathrm{d} F}{\mathrm{d} t}$ is a function of four variables: $x,y,\dot{x},\dot{y}$. So, the $\frac{\partial }{\partial x}$ in $\frac{\partial }{\partial x} \frac{\mathrm{d} F}{\mathrm{d} t}$ is not used in the same sense as equation (1)

Also, can I do something like this without reservation:

$$\frac{\mathrm{d^{10}} }{\mathrm{d^{10}} t} \frac{\partial^7 F}{\partial x^4 \partial y^3} = \frac{\partial^2}{\partial x \partial y} \frac{\mathrm{d^{2}} }{\mathrm{d^{2}} t} \frac{\partial^5 F}{\partial x^3 \partial y^2} \frac{\mathrm{d^{8}} F}{\mathrm{d^{8}} t}$$

$\frac{\partial }{\partial x}\frac{dF}{dt}$ has no sense... Don't forget that $$t\mapsto F(x(t),y(t))$$ is a function from $\mathbb R\to\mathbb R$.
• If I write $\frac{\mathrm{d} F}{\mathrm{d} t} = \frac{\partial F}{\partial x} \frac{\mathrm{d} x}{\mathrm{d} t} + \frac{\partial F}{\partial y} \frac{\mathrm{d} y}{\mathrm{d} t}$, I can express $\frac{\mathrm{d} F}{\mathrm{d} t}$ as a function of $x$. Would it then be possible to partially differentiate $\frac{\mathrm{d} F}{\mathrm{d} t}$ wrt $x$? – IanDsouza May 14 '15 at 6:28