(\partial)/(\partial x)((y/x)*ln(x))

解答

\frac{\partial}{\partial x} ((\frac{y}{x}) \cdot ln (x))

\frac{y ( - ln ( x ) + 1 )}{x ^{2}}

求解步骤

\frac{\partial}{\partial x} ((\frac{y}{x}) ln (x))

将

y

视为常数

将常数提出:

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

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

使用微分乘法定则:

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

f = \frac{1}{x}, g = ln (x)

= y (\frac{\partial}{\partial x} (\frac{1}{x}) ln (x) + \frac{\partial}{\partial x} (ln (x)) \frac{1}{x})

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

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

= y ((- \frac{1}{x ^{2}}) ln (x) + \frac{1}{x} \cdot \frac{1}{x})

化简

y ((- \frac{1}{x ^{2}}) ln (x) + \frac{1}{x} \cdot \frac{1}{x}) : \frac{y ( - ln ( x ) + 1 )}{x ^{2}}

= \frac{y ( - ln ( x ) + 1 )}{x ^{2}}

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

y^{^{''}} + 4 y^{^{'}} + 13 y = 0, y (0) = 3, y^{^{'}} (0) = - 12

x \to 2 lim (x - 5)

\int_{0}^{2} \frac{1}{( 2 x - 1 ) ^{\frac{2}{3}}} d x

\frac{\partial}{\partial x} (x^{4} + 3 y^{2} - x)