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What is implicit differentiation

  1. Jul 23, 2014 #1
    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 [itex]\frac{dy}{dx}[/itex] more than once, or functions of y, and application of the chain rule:

    [itex]\frac{df(y)}{dx}\,=\,f'(y) \frac{dy}{dx}[/itex] .

    Equations

    [tex]x^2\,+\,y^2\,=\,1[/tex] is an implicit definition of y.

    Its implicit derivative with respect to x is:

    [tex]2x\,+\,2y\frac{dy}{dx}\,=\,0[/tex]

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

    which in this case can be simplified to:

    [tex]\frac{dy}{dx}\,=\,-\frac{x}{y}[/tex]

    Extended explanation



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  2. jcsd
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