derivative of log_{10}((2x/(1+x^2)))

Lösung

\frac{d}{d x} (lo g_{10} (\frac{2 x}{1 + x ^{2}}))

\frac{1 - x ^{2}}{ln ( 10 ) x ( x ^{2} + 1 )}

Schritte zur Lösung

\frac{d}{d x} (lo g_{10} (\frac{2 x}{1 + x ^{2}}))

Wende die log Regel an:

lo g_{a} (b) = \frac{ln ( b )}{ln ( a )}

= \frac{d}{d x} (\frac{ln ( \frac{2 x}{1 + x ^{2}} )}{ln ( 10 )})

Entferne die Konstante:

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

= \frac{1}{ln ( 10 )} \frac{d}{d x} (ln (\frac{2 x}{1 + x ^{2}}))

Wende die Kettenregel an:

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

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

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

= \frac{1}{ln ( 10 )} \cdot \frac{1}{\frac{2 x}{1 + x ^{2}}} \cdot \frac{2 ( 1 - x ^{2} )}{( 1 + x ^{2} ) ^{2}}

Vereinfache

\frac{1}{ln ( 10 )} \cdot \frac{1}{\frac{2 x}{1 + x ^{2}}} \cdot \frac{2 ( 1 - x ^{2} )}{( 1 + x ^{2} ) ^{2}} : \frac{1 - x ^{2}}{ln ( 10 ) x ( x ^{2} + 1 )}

= \frac{1 - x ^{2}}{ln ( 10 ) x ( x ^{2} + 1 )}

\frac{d}{d x} (e^{a x^{4}})

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\frac{\partial}{\partial x} ((x - y)^{10})