$x^{2 }\+y^2\+z^2\=1$

$x^2\+y^2\+z^2\=1$

$\mathrm{implicitdiff}\left(\, z\left(\mathrm{x}\,\mathrm{y}\right)\, x\right)$

$-\frac{x}{z}$

$\mathrm{eval}\left(\,\mathrm{z}\=\mathrm{z}\left(\mathrm{x}\,\mathrm{y}\right)\right)$

$x^2\+y^2\+{z\left(x\,y\right)}^2\=1$

$\mathrm{diff}\left(\,x\right)$

$2x\+2z\left(x\,y\right)\left(\frac{\partial }{\partial x}z\left(x\,y\right)\right)\=0$

$\mathrm{isolate}\left(\, \mathrm{diff}\left(z\left(x\,y\right)\,x\right)\right)$$$

$\frac{\partial }{\partial x}z\left(x\,y\right)\=-\frac{x}{z\left(x\,y\right)}$

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

$z_x\=-\frac{x}{z}$