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Implicit Function Derivative

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Suppose that there is the dependence between argument х and function у given by the equation, unsolvable with regard to у. Such dependence defines у as the implicit function of х:

F(x;y)= 0. (*)

To obtain derivative у in variable х it is required to differentiate equation (*) in х and у, considering у as the composite function of х, i.e., multiplying by y¢х. In the obtained equation, we find the similar terms containing y¢х. And, solving it as the equation, we obtain the derivative y¢х.

Differentiation of the Function Given Parametrically

Suppose that there is function y(х) given parametrically:

Suppose that functions j(t) and y(t) are differentiable in parameter t and j¢t¹0, there is also the inverse function t=t-1(x). Then the derivative of function can be obtained by the formula: .


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