d/(dθ)((r(θ))(r(θ)sin(θ)))

解答

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

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

求解步骤

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

化简

(r (θ)) (r (θ) sin (θ)) : sin (θ) r (θ)^{2}

= \frac{d}{d θ} (sin (θ) r (θ)^{2})

使用微分乘法定则:

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

f = sin (θ), g = r (θ)^{2}

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

\frac{d}{d θ} (sin (θ)) = cos (θ)

\frac{d}{d θ} (r (θ)^{2}) = 2 r (θ) \frac{d}{d θ} (r (θ))

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

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