Implicit Differentiation
Implicit differentiation is the procedure of differentiating an implicit equation with respect to the desired variable x while treating the other variables as unspecified functions of x.
For example, the implicit equation
| xy=1 |
(1)
|
can be solved for
| [画像: y=1/x ] |
(2)
|
and differentiated directly to yield
| [画像: (dy)/(dx)=-1/(x^2). ] |
(3)
|
Differentiating implicitly instead gives
[画像:d/(dx)[xy]=d/(dx)[1] ]
![画像:d/(dx)[xy]=d/(dx)[1]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/ImplicitDifferentiation/Inline3.svg){kind=link}
(4)
[画像:x(dy)/(dx)+y(dx)/(dx)=0 ]
(5)
[画像:x(dy)/(dx)+y=0 ]
(6)
[画像:(dy)/(dx)=-y/x. ]
(7)
Plugging in y=1/x verifies that this approach gives the same result as before.
Implicit differentiation is especially useful when y^'(x) is needed, but it is difficult or inconvenient to solve for y in terms of x.
See also
Derivative, Differentiation Explore this topic in the MathWorld classroomExplore with Wolfram|Alpha
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References
Anton, H. "Implicit Differentiation." §3.6 in Calculus: A New Horizon, 6th ed. New York: Wiley, 1999.Referenced on Wolfram|Alpha
Implicit DifferentiationCite this as:
Weisstein, Eric W. "Implicit Differentiation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ImplicitDifferentiation.html