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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 х. In the obtained equation, we find the similar terms containing х. And, solving it as the equation, we obtain the derivative 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 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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