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

Lösung

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

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

Schritte zur Lösung

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

Behandle

y

als Konstante

Wende die Quotientenregel an:

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

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

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

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

x \to 0 lim ((1 + 5 x)^{\frac{1}{x}})

d er i v a t i v e f (x) = \frac{1}{1 - x}

\frac{d}{d x} (2 x^{- 6} + x^{- 1})

\int 8 e^{3 x} d x

n = 1 \sum \infty \frac{n - 1}{6 n - 1}