derivative of (ln(x)/(4x^2))

Lösung

\frac{d}{d x} (\frac{ln ( x )}{4 x ^{2}})

\frac{1 - 2 ln ( x )}{4 x ^{3}}

Schritte zur Lösung

\frac{d}{d x} (\frac{ln ( x )}{4 x ^{2}})

Entferne die Konstante:

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

= \frac{1}{4} \frac{d}{d x} (\frac{ln ( x )}{x ^{2}})

Wende die Quotientenregel an:

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

= \frac{1}{4} \cdot \frac{\frac{d}{d x} ( ln ( x ) ) x ^{2} - \frac{d}{d x} ( x ^{2} ) ln ( x )}{( x ^{2} ) ^{2}}

\frac{d}{d x} (ln (x)) = \frac{1}{x}

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

= \frac{1}{4} \cdot \frac{\frac{1}{x} x ^{2} - 2 x ln ( x )}{( x ^{2} ) ^{2}}

Vereinfache

\frac{1}{4} \cdot \frac{\frac{1}{x} x ^{2} - 2 x ln ( x )}{( x ^{2} ) ^{2}} : \frac{1 - 2 ln ( x )}{4 x ^{3}}

= \frac{1 - 2 ln ( x )}{4 x ^{3}}

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