derivative of (({g}(x)^2)/(h(x)))

Lösung

\frac{d}{d x} (\frac{( g ( x ) ) ^{2}}{h ( x )})

\frac{2 x g ( x ) \frac{d}{d x} ( g ( x ) ) - g ( x ) ^{2}}{x ^{2} h}

Schritte zur Lösung

\frac{d}{d x} (\frac{( g ( x ) ) ^{2}}{h ( x )})

Behandle

h

als Konstante

Entferne die Konstante:

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

= \frac{1}{h} \frac{d}{d x} (\frac{g ( x ) ^{2}}{x})

Wende die Quotientenregel an:

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

= \frac{1}{h} \cdot \frac{\frac{d}{d x} ( g ( x ) ^{2} ) x - \frac{d x}{d x} g ( x ) ^{2}}{x ^{2}}

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

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

= \frac{1}{h} \cdot \frac{2 g ( x ) \frac{d}{d x} ( g ( x ) ) x - 1 \cdot g ( x ) ^{2}}{x ^{2}}

Vereinfache

\frac{1}{h} \cdot \frac{2 g ( x ) \frac{d}{d x} ( g ( x ) ) x - 1 \cdot g ( x ) ^{2}}{x ^{2}} : \frac{2 x g ( x ) \frac{d}{d x} ( g ( x ) ) - g ( x ) ^{2}}{x ^{2} h}

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

\frac{d}{d x} (ln (\frac{2 - x}{2 + x}))

\int \frac{arctan ^{4} ( x )}{1 + x ^{2}} d x

x \to 0 lim (ln ((\frac{1}{x})^{x}))

\int (\frac{ln ( x )}{x ^{3}}) d x

\frac{d}{d x} (\frac{10}{( x - 2 ) ^{3}})