# Switching the order of differentiation

I have a problem with the problem below (the line with the red arrow)

Why is the first equality true, can you change the order of derivatives ? I mean is it allowed to write:

$$\dfrac{\partial}{\partial t}d_x\varphi_t=\dfrac{d}{dx}\bigg(\dfrac{\partial}{\partial t}\varphi_t\bigg)$$

does it follow from that ? then I could write:

$$\dfrac{d}{dx}\bigg(\dfrac{\partial}{\partial t}\varphi_t\bigg)=\dfrac{d}{dx}f(\varphi_t(x))=\dfrac{d}{d\varphi_t}f(\varphi_t(x))\cdot\dfrac{d}{dx}\varphi_t(x)=d_{\varphi_t(x)}fd_x\varphi_t$$

• "If the map [...] is sufficiently regular". For sufficiently regular maps, the mixed partial derivatives are independent of the order of differentiation. – Daniel Fischer Nov 4 '14 at 12:01
• @Daniel Fischer thanks a lot, didn't notice it. – inequal Nov 4 '14 at 12:37