Implicit differentiation with partial derivatives

When an equation mixes its variables together, it can be hard to solve for one of them before differentiating. We can get the derivative directly by moving everything to one side and taking partial derivatives. Let’s do that first for two variables, then for three.

Write the equation as F equal to zero

If $F (x, y) = 0$ and $y$ is a differentiable function of $x$ , the chain rule gives $F_{x} + F_{y} \frac{d y}{d x} = 0$ . Solve for the derivative wherever $F_{y} \neq 0$ .

\frac{d y}{d x} = - \frac{F_{x}}{F_{y}}

Find the derivative when x and y are mixed together

Let’s say $x e^{2 y} = x^{2} - y^{3}$ . Move everything to one side. When we take $F_{x}$ , we hold $y$ fixed. When we take $F_{y}$ , we hold $x$ fixed.

\begin{array}{l} F (x, y) = x^{2} - y^{3} - x e^{2 y} \\ F_{x} = 2 x - e^{2 y} \\ F_{y} = - 3 y^{2} - 2 x e^{2 y} \end{array}

\frac{d y}{d x} = \frac{2 x - e^{2 y}}{3 y^{2} + 2 x e^{2 y}}

Find the two partial derivatives of z

For $F (x, y, z) = 0$ with $z = z (x, y)$ , the same chain rule gives $z_{x} = - \frac{F_{x}}{F_{z}}$ and $z_{y} = - \frac{F_{y}}{F_{z}}$ wherever $F_{z} \neq 0$ . To find $z_{x}$ , hold $y$ fixed. To find $z_{y}$ , hold $x$ fixed.

For $x^{2} \sin z = 3 y z + 2 x^{3}$ , take the partial derivatives of $F$ first, treating $x$ , $y$ , and $z$ as independent inputs.

\begin{array}{l} F (x, y, z) = 3 y z + 2 x^{3} - x^{2} \sin z \\ F_{x} = 6 x^{2} - 2 x \sin z \\ F_{y} = 3 z \\ F_{z} = 3 y - x^{2} \cos z \end{array}

\begin{array}{l} z_{x} = - \frac{6 x^{2} - 2 x \sin z}{3 y - x^{2} \cos z} \\ z_{y} = - \frac{3 z}{3 y - x^{2} \cos z} \end{array}

Learn mathKrista KingAugust 30, 2020math, learn online, online course, online math, calculus 3, calculus iii, calc 3, calc iii, multivariable calculus, multivariable calc, multivariable functions, implicit differentiation, implicit derivatives, partial derivatives, partial differentiation

Implicit differentiation with partial derivatives

Write the equation as F equal to zero

Find the derivative when x and y are mixed together

Find the two partial derivatives of z

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