Question

Differentiate $\frac{\tan x}{x}\left(\log \frac{e^x}{x^x}\right)$.

Answer 1

Let $y=$$\frac{\tan x}{x}\left(\log \frac{e^x}{x^x}\right)$

or, $y=$$\frac{\tan x}{x}(\log e^x-\log x^x)$

or, $y=$$\frac{\tan x}{x}(x-x\log x)$

or, $y=\tan x-\tan x.\log x$

Now differentiating both sides with respect to $x$ we get,

$\frac{dy}{dx}={\sec }^{2}x-{\sec }^{2}x.\log x-\frac{\tan x}{x}$.