In
calculus,
implicit differentiation, an application of the
Chain Rule, allows one to
differentiate implicit functions. Consider
y +
x = -4. This function can be differentiated normally by using
algebra to change this
equation to an
explicit function:
f(
x) =
y = −
x - 4; such differentiation would result in a value of −1. Likewise, one can use implicit differentiation;
dy/
dx +
dx/
dx = 0 =
dy/
dx + 1;
dy/
dx = -1. Implicit differentiaton is used when the user cannot, or choses not to, use
algebra to manipulate an implicit function, until it becomes explicit.
An example of an implicit function, for which implicit differentiation might be easier than attempting to use explicit differentiation, is: x^{4} + 2y^{2} = 8. In order to explicitly differentiate this, one would have to obtain (via algebra) f(x) = {−[√(8 - x^{4})] / 2}, and then differentiate f(x). One might find it substantially easier to implicitly differentiate the implicit function; 4x^{3} + 4y(dy/dx) = 0; thus, dy/dx = −4x^{3} / 4y = −x^{3} / y.
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