derivative of xey(x^2+yln(x))

解

\frac{d}{d x} (x e y (x) (x)^{2} + y (x) ln (x))

e (3 x^{2} y (x) + \frac{d}{d x} (y (x)) x^{3}) + \frac{y ( x )}{x} + ln (x) \frac{d}{d x} (y (x))

解答ステップ

\frac{d}{d x} (x e y (x) (x)^{2} + y (x) ln (x))

簡素化

x e y (x) (x)^{2} + y (x) ln (x) : e x^{3} y (x) + ln (x) y (x)

= \frac{d}{d x} (e x^{3} y (x) + ln (x) y (x))

和/差の法則を適用:

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

= \frac{d}{d x} (e x^{3} y (x)) + \frac{d}{d x} (ln (x) y (x))

\frac{d}{d x} (e x^{3} y (x)) = e (3 x^{2} y (x) + \frac{d}{d x} (y (x)) x^{3})

\frac{d}{d x} (ln (x) y (x)) = \frac{y ( x )}{x} + ln (x) \frac{d}{d x} (y (x))

= e (3 x^{2} y (x) + \frac{d}{d x} (y (x)) x^{3}) + \frac{y ( x )}{x} + ln (x) \frac{d}{d x} (y (x))