If $f(x,y)= x^{x^{x^{x^y}}} + (\log x)(\arctan(\arctan(\arctan(\sin (\cos xy-\log (x+y)))))$ Find $D_2f(1,y)$

Question

Here we $D_2 f(1,y)$ means we have to calculate the partial derivative w.r.t $y$, so I have applied one short tricks that I have put $x=1$ in the equation then $f(1,y)= 1+0=1$ so the $D_2(f(1,y)=0$. Now my question is that the way I have gone is it correct or not. Please comment and give solution if I am wrong.

Answer 1

$$\frac{d}{dy}f(x,y)=x^{x^y+x^{x^{x^y}}+x^{x^y}+y} \log ^4(x)+\log{x} \frac{d}{dy}\Big(\arctan\left(\arctan\left(\arctan(\sin (\cos (x y)-\log (x+y)))\right)\right)\Big)$$

Hence $$\frac{d}{dy}f(x,y)\Big|_{x=1}=0.$$

Answer 2

Since $f(1,y)=1$ for all $y \neq 0$ (it is undefined at $y=0$), we have $$\left.\left(\frac{\partial}{\partial y}\,f(x,y)\right)\right|_{x=1}=\frac{\text{d}}{\text{d}y}\,f(1,y)=0\,.$$