derivative of (x^3-y^3/(x^2+y^2))

Lösung

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

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

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

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

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

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

Multipliziere aus

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

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

\frac{\partial}{\partial x} (cos^{4} (x^{8} y^{7}))

\int e^{- c o s (x)} sin (x) d x

\frac{\partial}{\partial x} (\frac{1}{4 x})

\frac{\partial}{\partial x} (3 x e^{7 x y})

\int_{0}^{7} \frac{4 t}{( t - 8 ) ^{2}} d t