derivative of (ax^2/(ln(x/4)))

解

\frac{d}{d x} (\frac{a x ^{2}}{ln ( \frac{x}{4} )})

\frac{a ( 2 x ln ( \frac{x}{4} ) - x )}{ln ^{2} ( \frac{x}{4} )}

解答ステップ

\frac{d}{d x} (\frac{a x ^{2}}{ln ( \frac{x}{4} )})

定数を除去:

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

= a \frac{d}{d x} (\frac{x ^{2}}{ln ( \frac{x}{4} )})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= a \frac{\frac{d}{d x} ( x ^{2} ) ln ( \frac{x}{4} ) - \frac{d}{d x} ( ln ( \frac{x}{4} ) ) x ^{2}}{( ln ( \frac{x}{4} ) ) ^{2}}

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

\frac{d}{d x} (ln (\frac{x}{4})) = \frac{1}{x}

= a \frac{2 x ln ( \frac{x}{4} ) - \frac{1}{x} x ^{2}}{( ln ( \frac{x}{4} ) ) ^{2}}

簡素化

a \frac{2 x ln ( \frac{x}{4} ) - \frac{1}{x} x ^{2}}{ln ^{2} ( \frac{x}{4} )} : \frac{a ( 2 x ln ( \frac{x}{4} ) - x )}{ln ^{2} ( \frac{x}{4} )}

= \frac{a ( 2 x ln ( \frac{x}{4} ) - x )}{ln ^{2} ( \frac{x}{4} )}