derivative of-(x^6/(sqrt(1+x^{12))})

Lösung

\frac{d}{d x} (- \frac{x ^{6}}{1 + x ^{12}})

- \frac{6 x ^{5}}{( 1 + x ^{12} ) 1 + x ^{12}}

Schritte zur Lösung

\frac{d}{d x} (- \frac{x ^{6}}{1 + x ^{12}})

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

= - \frac{d}{d x} (\frac{x ^{6}}{1 + x ^{12}})

Wende die Quotientenregel an:

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

= - \frac{\frac{d}{d x} ( x ^{6} ) 1 + x ^{12} - \frac{d}{d x} ( 1 + x ^{12} ) x ^{6}}{( 1 + x ^{12} ) ^{2}}

\frac{d}{d x} (x^{6}) = 6 x^{5}

\frac{d}{d x} (1 + x^{12}) = \frac{6 x ^{11}}{1 + x ^{12}}

= - \frac{6 x ^{5} 1 + x ^{12} - \frac{6 x ^{11}}{1 + x ^{12}} x ^{6}}{( 1 + x ^{12} ) ^{2}}

Vereinfache

- \frac{6 x ^{5} 1 + x ^{12} - \frac{6 x ^{11}}{1 + x ^{12}} x ^{6}}{( 1 + x ^{12} ) ^{2}} : - \frac{6 x ^{5}}{( 1 + x ^{12} ) 1 + x ^{12}}

= - \frac{6 x ^{5}}{( 1 + x ^{12} ) 1 + x ^{12}}

\int \frac{x}{( x + 3 ) ^{2}} d x

cot (x) y^{^{'}} = y

\frac{\partial}{\partial y} (e^{x^{2}} + y^{2})

x \to 3 + lim (\frac{x ^{2}}{x - 3})

\frac{d}{d x} (\frac{1}{x - 1} + \frac{1}{x + 1})