(\partial)/(\partial x)(xye^{sin(pixy)})

解答

\frac{\partial}{\partial x} (x y e^{s i n (π x y)})

y (e^{s i n (π x y)} + π e^{s i n (π x y)} x y cos (π x y))

求解步骤

\frac{\partial}{\partial x} (x y e^{s i n (π x y)})

将

y

视为常数

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= y \frac{\partial}{\partial x} (x e^{s i n (π x y)})

使用微分乘法定则:

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

f = x, g = e^{s i n (π x y)}

= y (\frac{\partial}{\partial x} (x) e^{s i n (π x y)} + \frac{\partial}{\partial x} (e^{s i n (π x y)}) x)

\frac{\partial}{\partial x} (x) = 1

\frac{\partial}{\partial x} (e^{s i n (π x y)}) = e^{s i n (π x y)} cos (π x y) π y

= y (1 \cdot e^{s i n (π x y)} + e^{s i n (π x y)} cos (π x y) π y x)

化简

= y (e^{s i n (π x y)} + π e^{s i n (π x y)} x y cos (π x y))

x \to - 3 lim ((5 - x) \frac{4}{3})

\int \frac{5}{1 + 9 x ^{2}} d x

\int y^{2} cosh (7 y) d y

x \to 0 lim (\frac{x}{4 ^{x} - 2 ^{x}})

\int_{0}^{7} \frac{8 t}{( t - 8 ) ^{2}} d t