Chapter 11 - Section 11.5 - Derivatives of Logarithmic and Exponential Functions - Exercises - Page 842: 49

$f^{\prime}(x)=e^{x}(\displaystyle \ln|x|+\frac{1}{x})$

f(x) is a product, $u(x)=e^{x},\quad v(x)=\ln|x|$ $f^{\prime}(x)=u(x)^{\prime}\cdot v(x)+u(x)\cdot v^{\prime}(x)$ $u^{\prime}(x)=e^{x},\quad v^{\prime}(x)=\displaystyle \frac{1}{x}$ $f^{\prime}(x)=e^{x}\displaystyle \ln|x|+e^{x}\cdot\frac{1}{x}$ $f^{\prime}(x)=e^{x}(\displaystyle \ln|x|+\frac{1}{x})$