derivative of ln((sqrt(x+3)/(e^{3x)}))

解

\frac{d}{d x} (ln (\frac{x + 3}{e ^{3 x}}))

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

解答ステップ

\frac{d}{d x} (ln (\frac{x + 3}{e ^{3 x}}))

簡素化

ln (\frac{x + 3}{e ^{3 x}}) : ln (x + 3) - 3 x

= \frac{d}{d x} (ln (x + 3) - 3 x)

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d x} (ln (x + 3)) - \frac{d}{d x} (3 x)

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

\frac{d}{d x} (3 x) = 3

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