d/(dt)(({z}(t,x,y))(x^2+y^2))

解

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

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

解答ステップ

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

積の法則を適用:

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

f = z (t, x, y), g = x^{2} + y^{2}

= \frac{d}{d t} (z (t, x, y)) (x^{2} + y^{2}) + \frac{d}{d t} (x^{2} + y^{2}) z (t, x, y)

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

= \frac{d}{d t} (z (t, x, y)) (x^{2} + y^{2}) + (2 x \frac{d x}{d t} + 2 y \frac{d y}{d t}) z (t, x, y)