derivative of e^{-x^2-y^2}cos(xy)

解

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

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

解答ステップ

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

積の法則を適用:

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

f = e^{- x^{2} - y^{2}}, g = cos (x y)

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

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

\frac{d}{d x} (cos (x y)) = - sin (x y) (y + x \frac{d y}{d x})

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

簡素化

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