derivative of (\sqrt[3]{x}/({f)(x)})

Soluzione

\frac{d}{d x} (\frac{3 x}{f ( x )})

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

Fasi della soluzione

\frac{d}{d x} (\frac{3 x}{f ( x )})

Applica la regola del quoziente:

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

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

\frac{d}{d x} (3 x) = \frac{1}{3 x ^{\frac{2}{3}}}

= \frac{\frac{1}{3 x ^{\frac{2}{3}}} f ( x ) - \frac{d}{d x} ( f ( x ) ) 3 x}{f ( x ) ^{2}}

Semplificare

\frac{\frac{1}{3 x ^{\frac{2}{3}}} f ( x ) - \frac{d}{d x} ( f ( x ) ) 3 x}{f ( x ) ^{2}} : \frac{f ( x ) - 3 3 x x ^{\frac{2}{3}} \frac{d}{d x} ( f ( x ) )}{3 x ^{\frac{2}{3}} f ( x ) ^{2}}

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

x \to 1 + lim (x^{\frac{1}{x - 1}})

im pl i c i t \frac{d y}{d x}, x y = 7

\int \frac{2 x ^{2}}{x ^{2} - 1} d x

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\int 2 x^{4} e^{x^{5}} d x