derivative of ({g}(x(sqrt(x)))/(ax^2+x))

Lösung

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

\frac{2 a x ^{2} x \frac{d}{d x} ( g ( x ) ) - 4 a x x g ( x ) + 2 x x \frac{d}{d x} ( g ( x ) ) + a x ^{\frac{3}{2}} g ( x ) - x g ( x )}{2 ( a x ^{2} + x ) ^{2}}

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

\frac{d}{d x} (g (x) x) = x \frac{d}{d x} (g (x)) + \frac{g ( x )}{2 x}

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

= \frac{( x \frac{d}{d x} ( g ( x ) ) + \frac{g ( x )}{2 x} ) ( a x ^{2} + x ) - ( 2 a x + 1 ) g ( x ) x}{( a x ^{2} + x ) ^{2}}

Vereinfache

\frac{( x \frac{d}{d x} ( g ( x ) ) + \frac{g ( x )}{2 x} ) ( a x ^{2} + x ) - ( 2 a x + 1 ) g ( x ) x}{( a x ^{2} + x ) ^{2}} : \frac{2 a x ^{2} x \frac{d}{d x} ( g ( x ) ) - 4 a x x g ( x ) + 2 x x \frac{d}{d x} ( g ( x ) ) + a x ^{\frac{3}{2}} g ( x ) - x g ( x )}{2 ( a x ^{2} + x ) ^{2}}

= \frac{2 a x ^{2} x \frac{d}{d x} ( g ( x ) ) - 4 a x x g ( x ) + 2 x x \frac{d}{d x} ( g ( x ) ) + a x ^{\frac{3}{2}} g ( x ) - x g ( x )}{2 ( a x ^{2} + x ) ^{2}}

t an g e n t \frac{7}{x ^{2} + 3}

\frac{d}{d x} (x^{2} sin (x) + 2 x cos (x))

\frac{d y}{d x} = \frac{2 y}{2 x + 1}, y (0) = e

\int_{0}^{4} x e^{x^{2}} d x

\int_{0}^{a} x x^{2} + a^{2} d x