Definition:Implicit Differentiation

Definition

Implicit differentiation is the differentiation of an implicit function with respect to the independent variable.

Examples

Arbitrary Example

Consider the implicit function:

$(1): \quad y^3 + 2 x^2 y + 8 = 0$

Using the technique of implicit differentiation, $(1)$ is differentiated with respect to $x$ thus:

$3 y^2 \dfrac {\d y} {\d x} + 4 x y + 2 x^2 \dfrac {\d y} {\d x} = 0$

from the Product Rule for Derivatives and the Power Rule for Derivatives.

This can then be algebraically manipulated to arrive at:

$\dfrac {\d y} {\d x} = -\dfrac {4 x y} {3 y^2 + 2 x^2}$

Also see

• Results about implicit differentiation can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): implicit differentiation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): implicit differentiation