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

解答

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

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

求解步骤

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

将

y

视为常数

使用连锁律:

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

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

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

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

化简

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

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

x \to 1 lim (4)

\int (2 x + 3)^{- 2} d x

y^{^{'}} + 3 y x^{5} = 0

n = 0 \sum \infty \frac{6}{4 ^{n}}

\int x e^{x + y} d x