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

解答

\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})

求解步骤

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

使用微分乘法定则:

(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}}

化简

= 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})

n = 1 \sum \infty n^{2} e^{- n^{3}}

in v erse l a pl a ce \frac{6}{s ^{2} + 9}

x \to 3 - lim (\frac{2}{( x - 3 )})

\frac{\partial}{\partial u ( v )} (\frac{e ^{v}}{u ( v ) + v ^{9}})

\int \frac{4 x}{4 + x} d x