derivative of ((x+3^2)/(e^x*cos(x)))

解

\frac{d}{d x} (\frac{( x + 3 ) ^{2}}{e ^{x} \cdot cos ( x )})

\frac{( 3 + x ) ( - cos ( x ) - x cos ( x ) + 3 sin ( x ) + x sin ( x ) )}{e ^{x} cos ^{2} ( x )}

解答ステップ

\frac{d}{d x} (\frac{( x + 3 ) ^{2}}{e ^{x} cos ( x )})

商の法則を適用:

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

= \frac{\frac{d}{d x} ( ( x + 3 ) ^{2} ) e ^{x} cos ( x ) - \frac{d}{d x} ( e ^{x} cos ( x ) ) ( x + 3 ) ^{2}}{( e ^{x} cos ( x ) ) ^{2}}

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

\frac{d}{d x} (e^{x} cos (x)) = e^{x} cos (x) - e^{x} sin (x)

= \frac{2 ( x + 3 ) e ^{x} cos ( x ) - ( e ^{x} cos ( x ) - e ^{x} sin ( x ) ) ( x + 3 ) ^{2}}{( e ^{x} cos ( x ) ) ^{2}}

簡素化

\frac{2 ( x + 3 ) e ^{x} cos ( x ) - ( e ^{x} cos ( x ) - e ^{x} sin ( x ) ) ( x + 3 ) ^{2}}{( e ^{x} cos ( x ) ) ^{2}} : \frac{( 3 + x ) ( - cos ( x ) - x cos ( x ) + 3 sin ( x ) + x sin ( x ) )}{e ^{x} cos ^{2} ( x )}

= \frac{( 3 + x ) ( - cos ( x ) - x cos ( x ) + 3 sin ( x ) + x sin ( x ) )}{e ^{x} cos ^{2} ( x )}