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

解答

\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)

求解步骤

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

使用微分乘法定则:

(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)

\frac{d}{d x} (\frac{sec ( x )}{7 + sec ( x )})

\frac{d}{d x} (ln (x + x^{2} - 8))

d er i v a t i v e \frac{ln ( x )}{x ^{7}}

d er i v a t i v e 7 x^{5} (x^{3} - 6 x)

d er i v a t i v e \frac{x - 6}{x - 6}