derivative of (x^2/(x+y))

解答

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

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

求解步骤

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

使用微分除法定则:

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

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

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

\frac{d}{d x} (x + y) = 1 + \frac{d y}{d x}

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

乘开

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

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

\frac{\partial}{\partial x} (x + y cos (x))

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

im pl i c i t \frac{d y}{d x}, y^{2} = x

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

d er i v a t i v e f (x) = 6 x