derivative of log_{10}(x^3+9)

Lösung

\frac{d}{d x} (lo g_{10} (x^{3} + 9))

\frac{3 x ^{2}}{ln ( 10 ) ( x ^{3} + 9 )}

Schritte zur Lösung

\frac{d}{d x} (lo g_{10} (x^{3} + 9))

Wende die log Regel an:

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

= \frac{d}{d x} (\frac{ln ( x ^{3} + 9 )}{ln ( 10 )})

Entferne die Konstante:

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

= \frac{1}{ln ( 10 )} \frac{d}{d x} (ln (x^{3} + 9))

Wende die Kettenregel an:

\frac{1}{x ^{3} + 9} \frac{d}{d x} (x^{3} + 9)

= \frac{1}{x ^{3} + 9} \frac{d}{d x} (x^{3} + 9)

\frac{d}{d x} (x^{3} + 9) = 3 x^{2}

= \frac{1}{ln ( 10 )} \cdot \frac{1}{x ^{3} + 9} \cdot 3 x^{2}

Vereinfache

\frac{1}{ln ( 10 )} \cdot \frac{1}{x ^{3} + 9} \cdot 3 x^{2} : \frac{3 x ^{2}}{ln ( 10 ) ( x ^{3} + 9 )}

= \frac{3 x ^{2}}{ln ( 10 ) ( x ^{3} + 9 )}