derivative of ax^2+by(x^2+λ(x^2+y^2-c^2))

解

\frac{d}{d x} (a x^{2} + b y (x) (x)^{2} + λ (x^{2} + y (x)^{2} - c^{2}))

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

解答ステップ

\frac{d}{d x} (a x^{2} + b y (x) (x)^{2} + λ (x^{2} + y (x)^{2} - c^{2}))

和/差の法則を適用:

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

= \frac{d}{d x} (a x^{2}) + \frac{d}{d x} (b y (x) x^{2}) + \frac{d}{d x} (λ (x^{2} + y (x)^{2} - c^{2}))

\frac{d}{d x} (a x^{2}) = 2 a x

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

\frac{d}{d x} (λ (x^{2} + y (x)^{2} - c^{2})) = λ (2 x + 2 y (x) \frac{d}{d x} (y (x)))

= 2 a x + b (\frac{d}{d x} (y (x)) x^{2} + 2 x y (x)) + λ (2 x + 2 y (x) \frac{d}{d x} (y (x)))