derivative of {r}(xe^{{r}(x)x})

解

\frac{d}{d x} (r (x) e^{r (x) x})

\frac{d}{d x} (r (x)) e^{r (x) x} + e^{r (x) x} (x \frac{d}{d x} (r (x)) + r (x)) r (x)

解答ステップ

\frac{d}{d x} (r (x) e^{r (x) x})

積の法則を適用:

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

f = r (x), g = e^{r (x) x}

= \frac{d}{d x} (r (x)) e^{r (x) x} + \frac{d}{d x} (e^{r (x) x}) r (x)

\frac{d}{d x} (e^{r (x) x}) = e^{r (x) x} (x \frac{d}{d x} (r (x)) + r (x))

= \frac{d}{d x} (r (x)) e^{r (x) x} + e^{r (x) x} (x \frac{d}{d x} (r (x)) + r (x)) r (x)