(\partial)/(\partial x)(xy(sin((x^5)+y^5)))

Lösung

\frac{\partial}{\partial x} (x y (sin ((x^{5}) + y^{5})))

y (sin (x^{5} + y^{5}) + 5 x^{5} cos (x^{5} + y^{5}))

Schritte zur Lösung

\frac{\partial}{\partial x} (x y (sin ((x^{5}) + y^{5})))

Behandle

y

als Konstante

Entferne die Konstante:

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

= y \frac{\partial}{\partial x} (x sin (x^{5} + y^{5}))

Wende die Produktregel an:

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

f = x, g = sin (x^{5} + y^{5})

= y (\frac{\partial}{\partial x} (x) sin (x^{5} + y^{5}) + \frac{\partial}{\partial x} (sin (x^{5} + y^{5})) x)

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

\frac{\partial}{\partial x} (sin (x^{5} + y^{5})) = cos (x^{5} + y^{5}) \cdot 5 x^{4}

= y (1 \cdot sin (x^{5} + y^{5}) + cos (x^{5} + y^{5}) \cdot 5 x^{4} x)

Vereinfache

y (1 \cdot sin (x^{5} + y^{5}) + cos (x^{5} + y^{5}) \cdot 5 x^{4} x) : y (sin (x^{5} + y^{5}) + 5 x^{5} cos (x^{5} + y^{5}))

= y (sin (x^{5} + y^{5}) + 5 x^{5} cos (x^{5} + y^{5}))