derivative f(x)=(x^2+sec(x))/(x-2e^x)

解

導関数

f (x) = \frac{x ^{2} + sec ( x )}{x - 2 e ^{x}}

\frac{x ^{2} + 2 e ^{x} x ^{2} - 4 e ^{x} x + x sec ( x ) tan ( x ) - 2 e ^{x} sec ( x ) tan ( x ) - sec ( x ) + 2 e ^{x} sec ( x )}{( x - 2 e ^{x} ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x ^{2} + sec ( x )}{x - 2 e ^{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 ^{2} + sec ( x ) ) ( x - 2 e ^{x} ) - \frac{d}{d x} ( x - 2 e ^{x} ) ( x ^{2} + sec ( x ) )}{( x - 2 e ^{x} ) ^{2}}

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

\frac{d}{d x} (x - 2 e^{x}) = 1 - 2 e^{x}

= \frac{( 2 x + sec ( x ) tan ( x ) ) ( x - 2 e ^{x} ) - ( 1 - 2 e ^{x} ) ( x ^{2} + sec ( x ) )}{( x - 2 e ^{x} ) ^{2}}

拡張

(2 x + sec (x) tan (x)) (x - 2 e^{x}) - (1 - 2 e^{x}) (x^{2} + sec (x)) : x^{2} + 2 e^{x} x^{2} - 4 e^{x} x + x sec (x) tan (x) - 2 e^{x} sec (x) tan (x) - sec (x) + 2 e^{x} sec (x)

= \frac{x ^{2} + 2 e ^{x} x ^{2} - 4 e ^{x} x + x sec ( x ) tan ( x ) - 2 e ^{x} sec ( x ) tan ( x ) - sec ( x ) + 2 e ^{x} sec ( x )}{( x - 2 e ^{x} ) ^{2}}