(\partial)/(\partial x)(3xarctan(x/y))

解答

\frac{\partial}{\partial x} (3 x arctan (\frac{x}{y}))

3 (arctan (\frac{x}{y}) + \frac{x y}{x ^{2} + y ^{2}})

求解步骤

\frac{\partial}{\partial x} (3 x arctan (\frac{x}{y}))

将

y

视为常数

将常数提出:

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

= 3 \frac{\partial}{\partial x} (x arctan (\frac{x}{y}))

使用微分乘法定则:

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

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

= 3 (\frac{\partial}{\partial x} (x) arctan (\frac{x}{y}) + \frac{\partial}{\partial x} (arctan (\frac{x}{y})) x)

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

\frac{\partial}{\partial x} (arctan (\frac{x}{y})) = \frac{y}{x ^{2} + y ^{2}}

= 3 (1 \cdot arctan (\frac{x}{y}) + \frac{y}{x ^{2} + y ^{2}} x)

化简

3 (1 \cdot arctan (\frac{x}{y}) + \frac{y}{x ^{2} + y ^{2}} x) : 3 (arctan (\frac{x}{y}) + \frac{x y}{x ^{2} + y ^{2}})

= 3 (arctan (\frac{x}{y}) + \frac{x y}{x ^{2} + y ^{2}})

d er i v a t i v e 4 e^{x} + x

t an g e n t (x + 1) (4 x^{2} - 4 x - 8)

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

t an g e n t f (x) = \frac{4}{x}, at x = 3

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