Implicit function theorem

In multivariable calculus, the implicit function theorem is a tool that allows relations to be converted to functions of several real variables. It does so by representing the relation as the graph of a function. There may not be a single function whose graph can represent the entire relation, but there may be such a function on a restriction of the domain of the relation. The implicit function theorem gives a sufficient condition to ensure that there is such a function.

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${\displaystyle f(a)-f(b)=\int _{b}^{a}f'(t)\,dt}$
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More precisely, given a system of m equations f_i (x₁, ..., x_n, y₁, ..., y_m) = 0, i = 1, ..., m (often abbreviated into F(x, y) = 0), the theorem states that, under a mild condition on the partial derivatives (with respect to each y_i ) at a point, the m variables y_i are differentiable functions of the x_j in some neighborhood of the point. As these functions can generally not be expressed in closed form, they are implicitly defined by the equations, and this motivated the name of the theorem.

In other words, under a mild condition on the partial derivatives, the set of zeros of a system of equations is locally the graph of a function.