d/(dθ)((r(θ))(sin(θ)+r(θ)cos(θ)))

Lösung

\frac{d}{d θ} ((r (θ)) (sin (θ) + r (θ) cos (θ)))

sin (θ) \frac{d}{d θ} (r (θ)) + 2 cos (θ) r (θ) \frac{d}{d θ} (r (θ)) + cos (θ) r (θ) - sin (θ) r (θ)^{2}

Schritte zur Lösung

\frac{d}{d θ} ((r (θ)) (sin (θ) + r (θ) cos (θ)))

Wende die Produktregel an:

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

f = r (θ), g = sin (θ) + r (θ) cos (θ)

= \frac{d}{d θ} (r (θ)) (sin (θ) + r (θ) cos (θ)) + \frac{d}{d θ} (sin (θ) + r (θ) cos (θ)) r (θ)

\frac{d}{d θ} (sin (θ) + r (θ) cos (θ)) = cos (θ) + cos (θ) \frac{d}{d θ} (r (θ)) - sin (θ) r (θ)

= \frac{d}{d θ} (r (θ)) (sin (θ) + r (θ) cos (θ)) + (cos (θ) + cos (θ) \frac{d}{d θ} (r (θ)) - sin (θ) r (θ)) r (θ)

Vereinfache

\frac{d}{d θ} (r (θ)) (sin (θ) + r (θ) cos (θ)) + (cos (θ) + cos (θ) \frac{d}{d θ} (r (θ)) - sin (θ) r (θ)) r (θ) : sin (θ) \frac{d}{d θ} (r (θ)) + 2 cos (θ) r (θ) \frac{d}{d θ} (r (θ)) + cos (θ) r (θ) - sin (θ) r (θ)^{2}

= sin (θ) \frac{d}{d θ} (r (θ)) + 2 cos (θ) r (θ) \frac{d}{d θ} (r (θ)) + cos (θ) r (θ) - sin (θ) r (θ)^{2}

e x p an d \frac{( p - 1 )}{( p ^{(2)} ( p ^{(2)} - 2 p + 5 ) )}

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

e x p an d \frac{5 s ^{2} + s + 4}{( s + 4 ) ( s ^{2} + 4 )}

e x p an d (6 - x + 2) (3 - x - 1)

e x p an d \frac{\partial}{\partial x} ((x - y)^{2}) + (y - 2)^{2}