I'm given the following equation:
$$ \frac{\partial \dot u}{\partial x} + \frac{\partial w}{\partial x}\frac{\partial \dot w}{\partial x} = 0 $$
where $u=u(x,t)$, $w=w(x,t)$ and $\dot{()} = \frac{\partial ()}{\partial t}$. My goal is to solve for $\frac{\partial \dot u}{\partial \dot w}$. One approach is to first solve for $\dot u$ as follows
$$ \dot u(x,t) = \int_0^x \frac{\partial^2 w}{\partial \xi^2} \dot w{\rm d}\xi - \left[ \frac{\partial w(\xi, t)}{\partial \xi} \dot w \right]_0^{x}+\dot u(0,t) $$
where integration-by-parts has been used. Now take the partial of both sides with respect to $\dot w$ to get
$$ \frac{\partial \dot u}{\partial \dot w} = \int_0^x \frac{\partial^2 w}{\partial \xi^2} {\rm d}{\xi} - \frac{\partial w}{\partial x} $$
Alternatively, I don't know why I can't simply multiply the first equation by $\left( \frac{\partial \dot w}{\partial x}\right)^{-1}$ to get
\begin{align} \frac{\partial x}{\partial \dot w}\frac{\partial \dot u}{\partial x} &= - \frac{\partial w}{\partial x}\frac{\partial \dot w}{\partial x}\frac{\partial x}{\partial \dot w} \\ \frac{\partial \dot u}{\partial \dot w} &= - \frac{\partial w}{\partial x} \end{align}
I'm assuming the first approach is correct and the second is incorrect, but I don't know why there is an error in the second. Can someone explain it to me?
