Editing Implicit function (section)

==== Example 3 ====
Often, it is difficult or impossible to solve explicitly for {{mvar|y}}, and implicit differentiation is the only feasible method of differentiation. An example is the equation

:<math>y^5-y=x \,.</math>

It is impossible to [[algebraic expression|algebraically express]] {{mvar|y}} explicitly as a function of {{mvar|x}}, and therefore one cannot find {{math|{{sfrac|''dy''|''dx''}}}} by explicit differentiation. Using the implicit method, {{math|{{sfrac|''dy''|''dx''}}}} can be obtained by differentiating the equation to obtain

:<math>5y^4\frac{dy}{dx} - \frac{dy}{dx} = \frac{dx}{dx} \,,</math>

where {{math|1={{sfrac|''dx''|''dx''}} = 1}}. Factoring out {{math|{{sfrac|''dy''|''dx''}}}} shows that

:<math>\left(5y^4 - 1\right)\frac{dy}{dx} = 1 \,,</math>

which yields the result

:<math>\frac{dy}{dx}=\frac{1}{5y^4-1} \,,</math>

which is defined for

:<math>y \ne \pm\frac{1}{\sqrt[4]{5}} \quad \text{and} \quad y \ne \pm \frac{i}{\sqrt[4]{5}} \,.</math>