<s>[INST] Question on partial derivative If $u= \log(\tan x + \tan y)$, prove that 
$$\sin(2x) \, \frac{du}{dx} + \sin(2y) \, \frac{du}{dy} = 2$$.
 [/INST] For $u = \ln(\tan x + \tan y)$ the derivatives with respect to $x$ and $y$ are
\begin{align}
\frac{du}{dx} &= \frac{sec^{2}(x)}{\tan x + \tan y} \\
\frac{du}{dy} &= \frac{sec^{2}(y)}{\tan x + \tan y}.
\end{align}
Now, 
\begin{align}
\sin(2x) \, \frac{du}{dx} + \sin(2y) \, \frac{du}{dy} &= 2 \, \frac{\tan x + \tan y}{\tan x + \tan y} = 2.
\end{align}
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