derivative y=xsqrt((a^2-x^2)/x)

Lösung

ableitung von

y = x \frac{a ^{2} - x ^{2}}{x}

\frac{a ^{2} - x ^{2}}{x} + \frac{- x ^{2} - a ^{2}}{2 x ^{\frac{1}{2}} - x ^{2} + a ^{2}}

Schritte zur Lösung

\frac{d}{d x} (x \frac{a ^{2} - x ^{2}}{x})

Wende die Produktregel an:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = \frac{a ^{2} - x ^{2}}{x}

= \frac{d x}{d x} \frac{a ^{2} - x ^{2}}{x} + \frac{d}{d x} (\frac{a ^{2} - x ^{2}}{x}) x

\frac{d x}{d x} = 1

\frac{d}{d x} (\frac{a ^{2} - x ^{2}}{x}) = \frac{- x ^{2} - a ^{2}}{2 x ^{\frac{3}{2}} - x ^{2} + a ^{2}}

= 1 \cdot \frac{a ^{2} - x ^{2}}{x} + \frac{- x ^{2} - a ^{2}}{2 x ^{\frac{3}{2}} - x ^{2} + a ^{2}} x

Vereinfache

1 \cdot \frac{a ^{2} - x ^{2}}{x} + \frac{- x ^{2} - a ^{2}}{2 x ^{\frac{3}{2}} - x ^{2} + a ^{2}} x : \frac{a ^{2} - x ^{2}}{x} + \frac{- x ^{2} - a ^{2}}{2 x ^{\frac{1}{2}} - x ^{2} + a ^{2}}

= \frac{a ^{2} - x ^{2}}{x} + \frac{- x ^{2} - a ^{2}}{2 x ^{\frac{1}{2}} - x ^{2} + a ^{2}}

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