### Multivariable Implicit Differentiation (Example 1)

Given $${x^2} + 2{y^2} + 3{z^2} = 1$$ find $$\frac{{\partial z}}{{\partial x}}$$ and $$\frac{{\partial z}}{{\partial y}}$$.

Let's find $$\frac{{\partial z}}{{\partial x}}$$ first. See how it's differentiation with respect to x? So differentiate x's normally. "z" is on top, so add $$\frac{{\partial z}}{{\partial x}}$$ after differentiating normally. "y" doesn't appear at all, so differentiating "y" will be the same as differentiating a constant, which results in 0.

${x^2} + 2{y^2} + 3{z^2} = 1$

Implicitly differentiating,

$2x + 0 + 6z\frac{{\partial z}}{{\partial x}} = 0$

And just rearrange,

$\frac{{\partial z}}{{\partial x}} = - \frac{{2x}}{{6z}}$

$\frac{{\partial z}}{{\partial x}} = - \frac{x}{{3z}}$

Done. Now let's do $$\frac{{\partial z}}{{\partial y}}$$. Now x is a constant, y is differentiated normally, and z is differentiated normally but you multiply by $$\frac{{\partial z}}{{\partial y}}$$.

${x^2} + 2{y^2} + 3{z^2} = 1$

$0 + 4y + 6z\frac{{\partial z}}{{\partial y}} = 0$

And rearrange,

$\frac{{\partial z}}{{\partial y}} = - \frac{{4y}}{{6z}}$

$\frac{{\partial z}}{{\partial y}} = - \frac{{2y}}{{3z}}$

Please see our other examples on partial derivatives, including chain rule, product rule, quotient rule, and multivariable implicit differentiation:

Partial Derivatives Examples:

Partial Derivatives (Ex1)

Partial Derivatives (Ex2)

Partial Derivatives (Ex3)

Partial Derivatives (Ex4)

Multivariable Chain Rule Examples:

Multivariable Chain Rule (Ex1)

Multivariable Chain Rule (Ex2) 