(\partial)/(\partial x)((y(xxxx-yyyy+4xxyy))/((xx+yy)^2))

Lösung

\frac{\partial}{\partial x} (\frac{y ( xxxx - yyyy + 4 xx yy )}{( xx + yy ) ^{2}})

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

Schritte zur Lösung

\frac{\partial}{\partial x} (\frac{y ( xxxx - yyyy + 4 xx yy )}{( xx + yy ) ^{2}})

Behandle

y

als Konstante

Entferne die Konstante:

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

= y \frac{\partial}{\partial x} (\frac{xxxx - yyyy + 4 xx yy}{( xx + yy ) ^{2}})

Wende die Quotientenregel an:

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

= y \frac{\frac{\partial}{\partial x} ( xxxx - yyyy + 4 xx yy ) ( xx + yy ) ^{2} - \frac{\partial}{\partial x} ( ( xx + yy ) ^{2} ) ( xxxx - yyyy + 4 xx yy )}{( ( xx + yy ) ^{2} ) ^{2}}

\frac{\partial}{\partial x} (xxxx - yyyy + 4 xx yy) = 4 x^{3} + 8 y^{2} x

\frac{\partial}{\partial x} ((xx + yy)^{2}) = 4 x (x^{2} + y^{2})

= y \frac{( 4 x ^{3} + 8 y ^{2} x ) ( xx + yy ) ^{2} - 4 x ( x ^{2} + y ^{2} ) ( xxxx - yyyy + 4 xx yy )}{( ( xx + yy ) ^{2} ) ^{2}}

Vereinfache

y \frac{( 4 x ^{3} + 8 y ^{2} x ) ( xx + yy ) ^{2} - 4 x ( x ^{2} + y ^{2} ) ( xxxx - yyyy + 4 xx yy )}{( ( xx + yy ) ^{2} ) ^{2}} : \frac{4 y ^{3} x ( 3 y ^{2} - x ^{2} )}{( x ^{2} + y ^{2} ) ^{3}}

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

x \to - 7 lim (\frac{x + 6}{x + 7})

\frac{d}{d x} ([arctan (x)] (1 + x^{2}))

\int \frac{x ^{2}}{( 1 - 16 x ^{2} ) ^{\frac{3}{2}}} d x

\int \frac{x ^{5} + 7}{x ^{2}} d x

\frac{d}{d x} (\frac{1}{4 + x ^{2}})