(\partial)/(\partial y)(x/(y^2+1))

解答

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

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

求解步骤

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

将

x

视为常数

将常数提出:

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

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

使用指数法则:

\frac{1}{a} = a^{- 1}

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

使用连锁律:

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

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

\frac{\partial}{\partial y} (y^{2} + 1) = 2 y

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

化简

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

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

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

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

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

d er i v a t i v e (1 + w) w^{3} + 9

d er i v a t i v e y = \frac{5 x}{x}