d/(dθ)(sqrt(l^2-(lsin(θ)-x(θ))^2)-lcos(θ))

解

\frac{d}{d θ} (l^{2} - (l sin (θ) - x (θ))^{2} - l cos (θ))

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

解答ステップ

\frac{d}{d θ} (l^{2} - (l sin (θ) - x (θ))^{2} - l cos (θ))

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d θ} (l^{2} - (l sin (θ) - x (θ))^{2}) - \frac{d}{d θ} (l cos (θ))

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

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

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

簡素化

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