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Implicit differentiation

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: x4 + 2y2 = 8. In order to explicitly differentiate this, one would have to obtain (via algebra) f(x) = {−[√(8 - x4)] / 2}, and then differentiate f(x). One might find it substantially easier to implicitly differentiate the implicit function; 4x3 + 4y(dy/dx) = 0; thus, dy/dx = −4x3 / 4y = −x3 / y.

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