derivative of e^{x/y}ln((x^2/y))

解

\frac{d}{d x} (e^{\frac{x}{y}} ln (\frac{x ^{2}}{y}))

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

解答ステップ

\frac{d}{d x} (e^{\frac{x}{y}} ln (\frac{x ^{2}}{y}))

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = e^{\frac{x}{y}}, g = ln (\frac{x ^{2}}{y})

= \frac{d}{d x} (e^{\frac{x}{y}}) ln (\frac{x ^{2}}{y}) + \frac{d}{d x} (ln (\frac{x ^{2}}{y})) e^{\frac{x}{y}}

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

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

= \frac{e ^{\frac{x}{y}} ( y - x \frac{d y}{d x} )}{y ^{2}} ln (\frac{x ^{2}}{y}) + \frac{2 y - x \frac{d y}{d x}}{x y} e^{\frac{x}{y}}

簡素化

\frac{e ^{\frac{x}{y}} ( y - x \frac{d y}{d x} )}{y ^{2}} ln (\frac{x ^{2}}{y}) + \frac{2 y - x \frac{d y}{d x}}{x y} e^{\frac{x}{y}} : \frac{e ^{\frac{x}{y}} ( y - x \frac{d y}{d x} ) ( 2 ln ( x ) - ln ( y ) )}{y ^{2}} + \frac{e ^{\frac{x}{y}} ( 2 y - x \frac{d y}{d x} )}{x y}

= \frac{e ^{\frac{x}{y}} ( y - x \frac{d y}{d x} ) ( 2 ln ( x ) - ln ( y ) )}{y ^{2}} + \frac{e ^{\frac{x}{y}} ( 2 y - x \frac{d y}{d x} )}{x y}