(\partial)/(\partial x)(xarcsin(y/x))

解答

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

arcsin (\frac{y}{x}) - \frac{y}{x ^{2} - y ^{2}}

求解步骤

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

将

y

视为常数

使用微分乘法定则:

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

f = x, g = arcsin (\frac{y}{x})

= \frac{\partial}{\partial x} (x) arcsin (\frac{y}{x}) + \frac{\partial}{\partial x} (arcsin (\frac{y}{x})) x

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

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

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

化简

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

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

d er i v a t i v e (x^{2} + 9)^{2} (x - 9)

\frac{d ^{5}}{d x ^{5}} (x^{\frac{11}{2}})

\int x + 1 d x

x \to 0 lim (\frac{4}{x ^{3} ( x + 3 )})

t a y l or sin (x), 1