derivative of-x^2sin(x^2y)

Lösung

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

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende die Produktregel an:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{2}, g = sin (x^{2} y)

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

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

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

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

Vereinfache

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

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

x \to 3 lim (x^{3} - 3 x^{2} - 9 x)

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

\int \frac{z ^{- 3}}{3} d z

y^{^{''}} + 16 y = 2 tan (4 x)

x \to - 1 lim (2 x)