derivative-(200ln(f/(40)))/(f+20)

解

導関数

- \frac{200 ln ( \frac{f}{40} )}{f + 20}

- \frac{200 ( f + 20 - f ln ( \frac{f}{40} ) )}{f ( f + 20 ) ^{2}}

解答ステップ

\frac{d}{df} - \frac{200 ln ( \frac{f}{40} )}{f + 20}

定数を除去:

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

= - 200 \frac{d}{df} \frac{ln ( \frac{f}{40} )}{f + 20}

商の法則を適用:

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

= - 200 \cdot \frac{\frac{d}{df} ( ln ( \frac{f}{40} ) ) ( f + 20 ) - \frac{d}{df} ( f + 20 ) ln ( \frac{f}{40} )}{( f + 20 ) ^{2}}

\frac{d}{df} (ln (\frac{f}{40})) = \frac{1}{f}

\frac{d}{df} (f + 20) = 1

= - 200 \cdot \frac{\frac{1}{f} ( f + 20 ) - 1 \cdot ln ( \frac{f}{40} )}{( f + 20 ) ^{2}}

簡素化

- 200 \cdot \frac{\frac{1}{f} ( f + 20 ) - 1 \cdot ln ( \frac{f}{40} )}{( f + 20 ) ^{2}} : - \frac{200 ( f + 20 - f ln ( \frac{f}{40} ) )}{f ( f + 20 ) ^{2}}

= - \frac{200 ( f + 20 - f ln ( \frac{f}{40} ) )}{f ( f + 20 ) ^{2}}