Implicit differentiation in multiple dimensions is a technique used to find partial derivatives when a relationship between variables is given in an implicit form rather than solved for one variable.

Process

1. Take the partial derivative of both sides with respect to the desired variable
2. Use the chain rule when necessary
3. Solve for the desired partial derivative within the equation

Example

Find given the equation of a sphere :

Generalized Equation

Consider a implicitly defined function , the partial derivative of any two inputs and can be defined as follows:

Example

Find given the equation of a sphere using the generalized equation: