(\partial)/(\partial θ)(r(θ)*sin(θ))

解

\frac{\partial}{\partial θ} (r (θ) \cdot sin (θ))

r (sin (θ) + θ cos (θ))

解答ステップ

\frac{\partial}{\partial θ} (r (θ) sin (θ))

r

を定数として扱う

定数を除去:

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

= r \frac{\partial}{\partial θ} (θ sin (θ))

積の法則を適用:

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

f = θ, g = sin (θ)

= r (\frac{\partial}{\partial θ} (θ) sin (θ) + \frac{\partial}{\partial θ} (sin (θ)) θ)

\frac{\partial}{\partial θ} (θ) = 1

\frac{\partial}{\partial θ} (sin (θ)) = cos (θ)

= r (1 \cdot sin (θ) + cos (θ) θ)

簡素化

= r (sin (θ) + θ cos (θ))