d/(dt)(t/(sqrt(1+t^2)))

Lösung

\frac{d}{d t} (\frac{t}{1 + t ^{2}})

\frac{1}{( 1 + t ^{2} ) 1 + t ^{2}}

Schritte zur Lösung

\frac{d}{d t} (\frac{t}{1 + t ^{2}})

Wende die Quotientenregel an:

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

= \frac{\frac{d t}{d t} 1 + t ^{2} - \frac{d}{d t} ( 1 + t ^{2} ) t}{( 1 + t ^{2} ) ^{2}}

\frac{d t}{d t} = 1

\frac{d}{d t} (1 + t^{2}) = \frac{t}{1 + t ^{2}}

= \frac{1 \cdot 1 + t ^{2} - \frac{t}{1 + t ^{2}} t}{( 1 + t ^{2} ) ^{2}}

Vereinfache

\frac{1 \cdot 1 + t ^{2} - \frac{t}{1 + t ^{2}} t}{( 1 + t ^{2} ) ^{2}} : \frac{1}{( 1 + t ^{2} ) 1 + t ^{2}}

= \frac{1}{( 1 + t ^{2} ) 1 + t ^{2}}

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