d/(dt)((e^{tb}-e^{ta})/(t(b-a)))

解

\frac{d}{d t} (\frac{e ^{t b} - e ^{t a}}{t ( b - a )})

\frac{b e ^{b t} t - a e ^{a t} t - e ^{b t} + e ^{a t}}{t ^{2} ( b - a )}

解答ステップ

\frac{d}{d t} (\frac{e ^{t b} - e ^{t a}}{t ( b - a )})

定数を除去:

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

= \frac{1}{b - a} \frac{d}{d t} (\frac{e ^{t b} - e ^{t a}}{t})

商の法則を適用:

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

= \frac{1}{b - a} \cdot \frac{\frac{d}{d t} ( e ^{t b} - e ^{t a} ) t - \frac{d t}{d t} ( e ^{t b} - e ^{t a} )}{t ^{2}}

\frac{d}{d t} (e^{t b} - e^{t a}) = e^{t b} b - e^{t a} a

\frac{d t}{d t} = 1

= \frac{1}{b - a} \cdot \frac{( e ^{t b} b - e ^{t a} a ) t - 1 \cdot ( e ^{t b} - e ^{t a} )}{t ^{2}}

簡素化

\frac{1}{b - a} \cdot \frac{( e ^{t b} b - e ^{t a} a ) t - 1 \cdot ( e ^{t b} - e ^{t a} )}{t ^{2}} : \frac{b e ^{b t} t - a e ^{a t} t - e ^{b t} + e ^{a t}}{t ^{2} ( b - a )}

= \frac{b e ^{b t} t - a e ^{a t} t - e ^{b t} + e ^{a t}}{t ^{2} ( b - a )}