Respuesta :

LammettHash

Use the quotient and chain rules.

Quotient rule:

[tex]\left(\dfrac{x}{\sqrt{1-x^2}}\right)' = \dfrac{(x)' \sqrt{1 - x^2} - x \left(\sqrt{1-x^2}\right)'}{\left(\sqrt{1-x^2}\right)^2}[/tex]

Chain rule:

[tex]\left(\sqrt{1-x^2}\right)' = \dfrac1{2\sqrt{1-x^2}} (1-x^2)' = -\dfrac x{\sqrt{1-x^2}}[/tex]

[tex]\implies \left(\dfrac{x}{\sqrt{1-x^2}}\right)' = \dfrac{\sqrt{1 - x^2} + \dfrac{x^2}{\sqrt{1-x^2}}}{1-x^2}[/tex]

Now just simplify.

[tex]\left(\dfrac{x}{\sqrt{1-x^2}}\right)' = \dfrac{\sqrt{1 - x^2} + \dfrac{x^2}{\sqrt{1-x^2}}}{1 - x^2} \\\\ ~~~~~~~~~~~~ = \dfrac1{\sqrt{1-x^2}} + \dfrac{x^2}{(1-x^2)^{3/2}} \\\\ ~~~~~~~~~~~~ = \dfrac1{\sqrt{1-x^2}} \left(1 + \dfrac{x^2}{1-x^2}\right) \\\\ ~~~~~~~~~~~~ = \dfrac1{\sqrt{1-x^2}} \cdot \dfrac1{1-x^2} \\\\ ~~~~~~~~~~~~ = \dfrac1{(1-x^2)^{3/2}}[/tex]

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