Question

$e^x\log \sqrt{x}\tan x$

Answer 1

$$\begin{gathered} \mathrm{Let}u=e^x;v=\log \sqrt{x};w=\tan x \\ \mathrm{Then},u\text{'}=e^x;v\text{'}=\frac{1}{\sqrt{x}}\times \frac{1}{2\sqrt{x}}=\frac{1}{2x};w\text{'}={\sec }^{2}x \\ \text{Using the product rule:} \\ \frac{d}{dx}\left(uvw\right)=u\text{'}vw+uv\text{'}w+uvw\text{'} \\ =e^x\log \sqrt{x}\tan x+e^x\times \frac{1}{2x}\tan x+e^x\log \sqrt{x}{\sec }^{2}x \\ =e^x\left(\log x^{\frac{1}{2}}.\tan x+\frac{\tan x}{2x}+\log x^{\frac{1}{2}}.{\sec }^{2}x\right) \\ =e^x\left(\frac{1}{2}\log x.\tan x+\frac{\tan x}{2x}+\frac{1}{2}\log x.{\sec }^{2}x\right) \\ =\frac{e^x}{2}\left(\log x.\tan x+\frac{\tan x}{x}+\log x.{\sec }^{2}x\right) \end{gathered}$$