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This is a noob calculus question.

(1) $\sqrt{|xy|} = \sqrt[4]{x^2y^2}$, are the 2 expressions equal?

if yes to (1),

then why $\frac{d}{dx}\sqrt{|xy|} \neq \frac{d}{dx}\sqrt[4]{x^2y^2}$ ?

as $\frac{d}{dx}\sqrt{|xy|} =\frac{\sqrt[4]{x^2y^2}}{2x} $

but $\frac{d}{dx}\sqrt[4]{x^2y^2} = \frac{0.5xy^2}{(x^2y^2)^{0.75}}$

i obtained the above derivatives from wolfram alpha.

-updated- yup, i think they are just 2 different way of presentation, seems like they have the same graph, thanks for the clarification.

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    Also, "are the 2 equations equal" is a little nonsensical. Maybe you mean "are the 2 expressions equal"?2011-11-10

2 Answers 2

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They are indeed equal. This follows since

$\begin{align} \sqrt[n]{x^n} = \left\{ \begin{array}{ccc} |x| & & n \text{ is even} \\ x & & n \text{ is odd} \end{array} \right. \end{align}$

The derivatives are indeed equal (although the derivatives of neither function exist at $0$).

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For (1)

Since $ \sqrt{|xy|} = \sqrt[4]{|xy|^2} $ and $|xy|=\pm xy $

$\sqrt{|xy|} = \sqrt[4]{(\pm xy)^2} = \sqrt[4]{x^2y^2} $