derivative of x/(sqrt(a^2-x^2))

Lösung

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

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

Schritte zur Lösung

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

Wende die Quotientenregel an:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

= \frac{1 \cdot a ^{2} - x ^{2} - ( - \frac{x}{a ^{2} - x ^{2}} ) x}{( a ^{2} - x ^{2} ) ^{2}}

Vereinfache

\frac{1 \cdot a ^{2} - x ^{2} - ( - \frac{x}{a ^{2} - x ^{2}} ) x}{( a ^{2} - x ^{2} ) ^{2}} : \frac{a ^{2}}{( a ^{2} - x ^{2} ) a ^{2} - x ^{2}}

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