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

Lösung

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

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

Schritte zur Lösung

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

Wende die log Regel an:

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

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

Entferne die Konstante:

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

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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