derivative of log_{4}(5x^2+1)

解

\frac{d}{d x} (lo g_{4} (5 x^{2} + 1))

\frac{5 x}{ln ( 2 ) ( 5 x ^{2} + 1 )}

解答ステップ

\frac{d}{d x} (lo g_{4} (5 x^{2} + 1))

対数の規則を適用する:

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

= \frac{d}{d x} (\frac{ln ( 5 x ^{2} + 1 )}{ln ( 4 )})

定数を除去:

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

= \frac{1}{ln ( 4 )} \frac{d}{d x} (ln (5 x^{2} + 1))

連鎖法則を適用:

\frac{1}{5 x ^{2} + 1} \frac{d}{d x} (5 x^{2} + 1)

= \frac{1}{5 x ^{2} + 1} \frac{d}{d x} (5 x^{2} + 1)

\frac{d}{d x} (5 x^{2} + 1) = 10 x

= \frac{1}{ln ( 4 )} \cdot \frac{1}{5 x ^{2} + 1} \cdot 10 x

簡素化

\frac{1}{ln ( 4 )} \cdot \frac{1}{5 x ^{2} + 1} \cdot 10 x : \frac{5 x}{ln ( 2 ) ( 5 x ^{2} + 1 )}

= \frac{5 x}{ln ( 2 ) ( 5 x ^{2} + 1 )}