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

Lösung

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

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

Schritte zur Lösung

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

Wende die Summen-/Differenzregel an:

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

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

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

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

\frac{d}{d x} (3 x) = 3

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

Vereinfache

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

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