Obtain $u_x$ from first principles

$u\left(x\,y\,z\right)\=x\+y\+z\left(x\,y\right)\+{\ⅇ}^{u\left(x\,y\,z\right)}$

$\stackrel{\text{differentiate w.r.t. x}}{\to }$

$z\left(x\,y\right)\=x\+y\+\cos \left(z\left(x\,y\right)\right)$

$\stackrel{\text{differentiate w.r.t. x}}{\to }$

$\,$

$\stackrel{\text{solve (specified)}}{\to }$

Obtain $u_y$ from first principles

$u\left(x\,y\,z\right)\=x\+y\+z\left(x\,y\right)\+{\ⅇ}^{u\left(x\,y\,z\right)}$

$\stackrel{\text{differentiate w.r.t. y}}{\to }$

$z\left(x\,y\right)\=x\+y\+\cos \left(z\left(x\,y\right)\right)$

$\stackrel{\text{differentiate w.r.t. y}}{\to }$

$\,$

$\stackrel{\text{solve (specified)}}{\to }$

Apply the implicitdiff command

$\mathrm{implicitdiff}\left(\left\{u\=x\+y\+z\+{\ⅇ}^u\,z\=x\+y\+\cos \left(z\right)\right\}\,\left\{u\,z\right\}\,\left\{u\,z\right\}\,x\right)$

$\left\{{\mathrm{D}}_1\left(u\right)\=-\frac{2\+\sin \left(z\right)}{\left({\ⅇ}^u-1\right)\left(\sin \left(z\right)\+1\right)}\,{\mathrm{D}}_1\left(z\right)\=\frac{1}{\sin \left(z\right)\+1}\right\}$

$\mathrm{implicitdiff}\left(\left\{u\=x\+y\+z\+{\ⅇ}^u\,z\=x\+y\+\cos \left(z\right)\right\}\,\left\{u\,z\right\}\,\left\{u\,z\right\}\,y\right)$

$\left\{{\mathrm{D}}_2\left(u\right)\=-\frac{2\+\sin \left(z\right)}{\left({\ⅇ}^u-1\right)\left(\sin \left(z\right)\+1\right)}\,{\mathrm{D}}_2\left(z\right)\=\frac{1}{\sin \left(z\right)\+1}\right\}$

Notational simplifications

$$\begin{gathered} \mathrm{interface}\left(\mathrm{typesetting}\=\mathrm{extended}\right)\: \\ \mathrm{Typesetting}:-\mathrm{Suppress}\left(u\left(x\,y\,z\right)\right)\: \\ \mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}\=\mathrm{true}\right)\: \end{gathered}$$

Obtain $u_x$ 

$$\begin{gathered} q_1≔u\left(x\,y\,z\right)\=x\+y\+z\left(x\,y\right)\+{\ⅇ}^{u\left(x\,y\,z\right)}\; \\ {q}_2≔z\left(x\,y\right)\=x\+y\+\cos \left(z\left(x\,y\right)\right) \end{gathered}$$

$q_1≔u=x+y+z\left(x\,y\right)+{\ⅇ}^u$

$q_3≔\mathrm{solve}\left(\left[\mathrm{diff}\left(q_1\,x\right)\,\mathrm{diff}\left(q_2\,x\right)\right]\,\left[\mathrm{diff}\left(u\,x\right)\,\mathrm{diff}\left(z\left(x\,y\right)\,x\right)\right]\right)$

$q_3≔\left[\left[u_x=-\frac{2+\sin \left(z\left(x\,y\right)\right)}{\left({\ⅇ}^u-1\right)\left(\sin \left(z\left(x\,y\right)\right)+1\right)}\,\frac{\partial }{\partial x}z\left(x\,y\right)=\frac{1}{\sin \left(z\left(x\,y\right)\right)+1}\right]\right]$

$\mathrm{eval}\left(\mathrm{op}\left(\left[1\,1\right]\,q_3\right)\,\left[z\left(x\,y\right)\=z\right]\right)$

$u_x=-\frac{2+\sin \left(z\right)}{\left({\ⅇ}^u-1\right)\left(\sin \left(z\right)+1\right)}$

Obtain $u_y$ 

$q_4≔\mathrm{solve}\left(\left[\mathrm{diff}\left(q_1\,y\right)\,\mathrm{diff}\left(q_2\,y\right)\right]\,\left[\mathrm{diff}\left(u\,y\right)\,\mathrm{diff}\left(z\left(x\,y\right)\,y\right)\right]\right)$

$q_4≔\left[\left[u_y=-\frac{2+\sin \left(z\left(x\,y\right)\right)}{\left({\ⅇ}^u-1\right)\left(\sin \left(z\left(x\,y\right)\right)+1\right)}\,\frac{\partial }{\partial y}z\left(x\,y\right)=\frac{1}{\sin \left(z\left(x\,y\right)\right)+1}\right]\right]$

$\mathrm{eval}\left(\mathrm{op}\left(\left[1\,1\right]\,q_4\right)\,\left[z\left(x\,y\right)\=z\right]\right)$

$u_y=-\frac{2+\sin \left(z\right)}{\left({\ⅇ}^u-1\right)\left(\sin \left(z\right)+1\right)}$