d/(dt)(({z}(t,x,y))(xy(t)^5-x^2y))

解

\frac{d}{d t} ((z (t, x, y)) (x y (t) (t)^{5} - x^{2} y (t)))

\frac{d}{d t} (z (t, x, y)) (x y (t) t^{5} - x^{2} y (t)) + (t^{5} \frac{d x}{d t} y (t) + t^{5} x \frac{d}{d t} (y (t)) + 5 t^{4} x y (t) - 2 x \frac{d x}{d t} y (t) - \frac{d}{d t} (y (t)) x^{2}) z (t, x, y)

解答ステップ

\frac{d}{d t} ((z (t, x, y)) (x y (t) (t)^{5} - x^{2} y (t)))

積の法則を適用:

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

f = z (t, x, y), g = x y (t) t^{5} - x^{2} y (t)

= \frac{d}{d t} (z (t, x, y)) (x y (t) t^{5} - x^{2} y (t)) + \frac{d}{d t} (x y (t) t^{5} - x^{2} y (t)) z (t, x, y)

\frac{d}{d t} (x y (t) t^{5} - x^{2} y (t)) = t^{5} \frac{d x}{d t} y (t) + t^{5} x \frac{d}{d t} (y (t)) + 5 t^{4} x y (t) - 2 x \frac{d x}{d t} y (t) - \frac{d}{d t} (y (t)) x^{2}

= \frac{d}{d t} (z (t, x, y)) (x y (t) t^{5} - x^{2} y (t)) + (t^{5} \frac{d x}{d t} y (t) + t^{5} x \frac{d}{d t} (y (t)) + 5 t^{4} x y (t) - 2 x \frac{d x}{d t} y (t) - \frac{d}{d t} (y (t)) x^{2}) z (t, x, y)