What is implicit differentiation

[Greg Bernhardt]

Definition/Summary

The definition of a function y of x is explicit if it is an equation in which y appears only once, and on its own (usually by starting "y =").

In any other case, the definition of a function y of x is implicit.

Implicit differentiation of y with respect to x is a slightly misleading name for ordinary differentiation of the defining equation of y.

Therefore, it generally involves $\frac{dy}{dx}$ more than once, or functions of y, and application of the chain rule:

$\frac{df(y)}{dx}\,=\,f'(y) \frac{dy}{dx}$ .

Equations

$$x^2\,+\,y^2\,=\,1$$ is an implicit definition of y.

Its implicit derivative with respect to x is:

$$2x\,+\,2y\frac{dy}{dx}\,=\,0$$

(where the chain rule has been applied by differentiating $y^2$ with respect to y, and then multiplying by $\frac{dy}{dx}$)

which in this case can be simplified to:

$$\frac{dy}{dx}\,=\,-\frac{x}{y}$$

Extended explanation



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[fresh_42]

The implicit function theorem is one of the most important theorems in calculus.
See https://en.wikipedia.org/wiki/Implicit_function_theorem