derivative of y/(y+x^2)

解答

\frac{d}{d x} (\frac{y}{y + x ^{2}})

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

求解步骤

\frac{d}{d x} (\frac{y}{y + x ^{2}})

使用微分除法定则:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

乘开

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

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

\frac{d}{d x} (e^{x^{2}} sin (x))

\frac{d x}{d t} = 2 x (10 - x)

\frac{\partial}{\partial t} (e^{t} sin (t))

\frac{d}{d x} (3 ln (4 x))

\int_{1}^{3} (4 x - 3)^{2} - (x^{2})^{2} d x