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)

Return to the multivariable differentiation hub page:

Partial Derivatives, Multivariable Differentiation

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