(\partial)/(\partial x)(6x^3cos(6x^2y))

Lösung

\frac{\partial}{\partial x} (6 x^{3} cos (6 x^{2} y))

6 (3 x^{2} cos (6 x^{2} y) - 12 x^{4} y sin (6 x^{2} y))

Schritte zur Lösung

\frac{\partial}{\partial x} (6 x^{3} cos (6 x^{2} y))

Behandle

y

als Konstante

Entferne die Konstante:

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

= 6 \frac{\partial}{\partial x} (x^{3} cos (6 x^{2} y))

Wende die Produktregel an:

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

f = x^{3}, g = cos (6 x^{2} y)

= 6 (\frac{\partial}{\partial x} (x^{3}) cos (6 x^{2} y) + \frac{\partial}{\partial x} (cos (6 x^{2} y)) x^{3})

\frac{\partial}{\partial x} (x^{3}) = 3 x^{2}

\frac{\partial}{\partial x} (cos (6 x^{2} y)) = - sin (6 x^{2} y) \cdot 12 y x

= 6 (3 x^{2} cos (6 x^{2} y) + (- sin (6 x^{2} y) \cdot 12 y x) x^{3})

Vereinfache

6 (3 x^{2} cos (6 x^{2} y) + (- sin (6 x^{2} y) \cdot 12 y x) x^{3}) : 6 (3 x^{2} cos (6 x^{2} y) - 12 x^{4} y sin (6 x^{2} y))

= 6 (3 x^{2} cos (6 x^{2} y) - 12 x^{4} y sin (6 x^{2} y))