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Beliebt Rechnen >

derivative of log_{2}(((3x+1)/(x^2+1)))

Lösung

\frac{d}{d x} (lo g_{2} (\frac{( 3 x + 1 )}{x ^{2} + 1}))

Lösung

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

Schritte zur Lösung

\frac{d}{d x} (lo g_{2} (\frac{( 3 x + 1 )}{x ^{2} + 1}))

Wende die log Regel an:

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

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

Entferne die Konstante:

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

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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

Graph

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