derivative of 4xy^2-xy^3-x^2y^2

Lösung

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

4 y^{2} + 8 x y \frac{d y}{d x} - y^{3} - 3 x y^{2} \frac{d y}{d x} - 2 x y^{2} - 2 x^{2} y \frac{d y}{d x}

Schritte zur Lösung

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

Wende die Summen-/Differenzregel an:

(f \pm g)^{'} = f^{'} \pm g^{'}

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

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

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

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

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

Vereinfache

4 (y^{2} + 2 x y \frac{d y}{d x}) - (y^{3} + 3 x y^{2} \frac{d y}{d x}) - (2 x y^{2} + 2 y \frac{d y}{d x} x^{2}) : 4 y^{2} + 8 x y \frac{d y}{d x} - y^{3} - 3 x y^{2} \frac{d y}{d x} - 2 x y^{2} - 2 x^{2} y \frac{d y}{d x}

= 4 y^{2} + 8 x y \frac{d y}{d x} - y^{3} - 3 x y^{2} \frac{d y}{d x} - 2 x y^{2} - 2 x^{2} y \frac{d y}{d x}

d er i v a t i v e cot^{2} (π x)

d er i v a t i v e 8 (2 z e^{z} + e^{z} z^{2})

\frac{d y}{d x} = - 4 x y^{2}

x \to - 3 lim (x + 1)

a re a x + 2, \frac{1}{x + 1}, x = 2