Implicit differentiation

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

$F(x,y)=0$

defines $y$ as a function of $x$, at least locally, implicit differentiation treats $y$ as a function $y(x)$ and differentiates both sides of the equation with respect to $x$. 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 terms. For example, the circle $x^{2}+y^{2}=1$ cannot be represented globally as the graph of a single function $y=f(x)$, but its upper and lower arcs can each be differentiated implicitly.

The method allows for the computation of the tangent line approximation to $y(x)$, given only the function $F$ and the point $(x_{0},y_{0})$ at which the approximation is made, so that no other knowledge of the functional dependence of $y$ on $x$ is needed. Tangent approximations to surfaces and higher-dimensional manifolds given by an equation or system of equations can be found by similar methods.