Write the first equation with the appropriate dependencies made explicit.

Context Panel: Differentiate≻With Respect To≻$x$

$u\left(x\,y\,z\right)\=x^3\+y^3\+{z\left(x\,y\right)}^3\+{u\left(x\,y\,z\right)}^3$

$\frac{\partial }{\partial x}u\left(x\,y\,z\right)\=3x^2\+3{z\left(x\,y\right)}^2\left(\frac{\partial }{\partial x}z\left(x\,y\right)\right)\+3{u\left(x\,y\,z\right)}^2\left(\frac{\partial }{\partial x}u\left(x\,y\,z\right)\right)$

Write the second equation with the appropriate dependencies made explicit.

Context Panel: Differentiate≻With Respect To≻$x$ 

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

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

$\frac{\partial }{\partial x}u\left(x\,y\,z\right)\=3x^2\+3{z\left(x\,y\right)}^2\left(\frac{\partial }{\partial x}z\left(x\,y\right)\right)\+3{u\left(x\,y\,z\right)}^2\left(\frac{\partial }{\partial x}u\left(x\,y\,z\right)\right)\,\frac{\partial }{\partial x}z\left(x\,y\right)\=2x\+2z\left(x\,y\right)\left(\frac{\partial }{\partial x}z\left(x\,y\right)\right)$

$\left\{\frac{\partial }{\partial x}u\left(x\,y\,z\right)\=\frac{3x\left(2{z\left(x\,y\right)}^2-2z\left(x\,y\right)x\+x\right)}{\left(2z\left(x\,y\right)-1\right)\left(3{u\left(x\,y\,z\right)}^2-1\right)}\,\frac{\partial }{\partial x}z\left(x\,y\right)\=-\frac{2x}{2z\left(x\,y\right)-1}\right\}$

Write the first equation with the appropriate dependencies made explicit.

Context Panel: Differentiate≻With Respect To≻$y$

$u\left(x\,y\,z\right)\=x^3\+y^3\+{z\left(x\,y\right)}^3\+{u\left(x\,y\,z\right)}^3$

$\frac{\partial }{\partial y}u\left(x\,y\,z\right)\=3y^2\+3{z\left(x\,y\right)}^2\left(\frac{\partial }{\partial y}z\left(x\,y\right)\right)\+3{u\left(x\,y\,z\right)}^2\left(\frac{\partial }{\partial y}u\left(x\,y\,z\right)\right)$

Write the second equation with the appropriate dependencies made explicit.

Context Panel: Differentiate≻With Respect To≻$y$ 

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

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

$\frac{\partial }{\partial y}u\left(x\,y\,z\right)\=3y^2\+3{z\left(x\,y\right)}^2\left(\frac{\partial }{\partial y}z\left(x\,y\right)\right)\+3{u\left(x\,y\,z\right)}^2\left(\frac{\partial }{\partial y}u\left(x\,y\,z\right)\right)\,\frac{\partial }{\partial y}z\left(x\,y\right)\=2y\+2z\left(x\,y\right)\left(\frac{\partial }{\partial y}z\left(x\,y\right)\right)$

$\left\{\frac{\partial }{\partial y}u\left(x\,y\,z\right)\=\frac{3y\left(2{z\left(x\,y\right)}^2-2z\left(x\,y\right)y\+y\right)}{\left(2z\left(x\,y\right)-1\right)\left(3{u\left(x\,y\,z\right)}^2-1\right)}\,\frac{\partial }{\partial y}z\left(x\,y\right)\=-\frac{2y}{2z\left(x\,y\right)-1}\right\}$

These commands cause $u$ and $u\left(x\,y\,z\right)$ to be equivalent, and for its derivatives to be written with subscripts.

Write the two defining equations with the appropriate dependencies explicitly stated.
Press the Enter key.

Differentiate both equations with respect to $x$ and solve for the two derivatives $u_x$ and $z_x$.

Extract the solution for $u_x$ and replace $z\left(x\,y\right)$ with $z$.

Differentiate both equations with respect to $y$ and solve for the two derivatives $u_y$ and $z_y$.

Extract the solution for $u_y$ and replace $z\left(x\,y\right)$ with $z$.