In calculus, implicit differentiation is a method for finding the derivative of a function that is defined by an equation rather than by an explicit formula. If an equation such as

. The method is an application of the chain rule.

Implicit differentiation is useful when solving explicitly for one variable is inconvenient, produces several branches, or is not possible in elementary, but its upper and lower arcs can each be differentiated implicitly.

The method allows for the computation of the tangent line approximation is needed. Tangent approximations to surfaces and higher-dimensional manifolds given by an equation or system of equations can be found by similar methods.

Basic formula

be a differentiable function of two variables, and suppose that an equation

. Applying the differential on the curve of interest, and writing the left-hand side in terms of the partial derivatives gives.

.

Relation with the implicit function theorem

at all.

. While this fact allows one to compute a numerical value, it is not obvious that the numerical value has the advertised meaning: namely, that it be the derivative of a function through the point.

The implicit function theorem.

Examples

Example 1

Consider

This equation is easy to solve for , giving

where the right side is the explicit form of the function . Differentiation then gives .

Alternatively, one can totally differentiate the original equation:

Solving for gives

the same answer as obtained previously.

Example 2

An example of an implicit function for which implicit differentiation is easier than using explicit differentiation is the function defined by the equation

To differentiate this explicitly with respect to , one has first to get

and then differentiate this function. This creates two derivatives: one for and another for .

It is substantially easier to implicitly differentiate the original equation:

giving

Example 3

Often, it is difficult or impossible to solve explicitly for , and implicit differentiation is the only feasible method of differentiation. An example is the equation

It is impossible to express explicitly in radicals as a function of , and therefore one cannot find by explicit differentiation of elementary radical expressions. Using the implicit method, can be obtained by differentiating the equation to obtain

where . Factoring out shows that

which yields the result

which is defined for

Higher order derivatives

The method of implicit differentiation can be applied to find higher-order derivatives such as the second derivative is sufficiently smooth at the point.

.

. Also, substituting the known exist and are continuous in a disc around the point.

In that case, Clairaut's theorem

Higher order derviatives (under the requisite smoothness hypotheses) are handled similarly.

References