Chapter 4.4, Problem 44E

To determine

To find $\frac{dy}{dx}$ of the given function by using logarithmic differentiation.

Answer to Problem 44E

  $\frac{dy}{dx}$ of the function $y=x^{\tan x},x>0$ is $\frac{dy}{dx}=x^{\tan x}\left({\sec }^{2}x\left(\ln x\right)+\frac{\tan x}{x}\right)$ when $x=0$ it is indeterminate.

Explanation of Solution

Given information:

  $y=x^{\tan x},x>0$

Calculation:

  $\begin{array}{c}\ln y=\tan x\ln x\\ \frac{1}{y}\frac{dy}{dx}=\tan x\times \frac{1}{x}+\ln x\times \ln {\sec }^{2}x\\ \frac{dy}{dx}=x^{\tan x}\left(\frac{\tan x}{x}+\ln x{\sec }^{2}x\right)\end{array}$

At $x=0$ ,

  $0^{\tan 0}\left({\sec }^{2}0\left(\ln 0\right)+\frac{\tan 0}{0}\right)=\text{indeterminate}$

Thus we can find $\frac{dy}{dx}$ of the given function.