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

Lösung

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

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

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

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

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

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

Vereinfache

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

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

a re a y = 4 - x^{2}, y = x + 2

\frac{\partial}{\partial y} (\frac{1}{x y})

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

\int \frac{3}{x ^{2}} - \frac{2}{x ^{3}} d x

t an g e n t x^{3}