Test Index

AP EAMCET Engineering 21 Apr 2019 Shift 1 Paper

Section: Mathematics

© examsnet.com

Question : 66 of 160

Marks: +1, -0

if f (x) =

\cot^{-1}\left(\frac{x^x-x^{-x}}{2}\right)

, then f' (1) =

- log 2
log 2
1
- 1

Solution:

Consider the function.

f(x)=\cot^{-1}\left(\;\frac{x^{x}-x^{-x}}{2}\right)

Let,

y=\cot^{-1}\left(\;\frac{x^{2x}-1}{2x^{x}}\right)

Put

x^{x}=\tan\theta

then

y\;=\cot^{-1}\left(\;\frac{\tan^{2}\theta-1}{2\tan\theta}\right)

\;\;=\cot^{-1}(-\cot 2\theta)

\;\;=\pi-\cot^{-1}(\cot 2\theta)

\;\;=\pi-\theta

This implies

y=\pi-2\tan^{-1}(x^{x})

Differentiate above,

\frac{dy}{dx}=\;\frac{-2}{1+x^{2x}}\cdot x^{x}(1+\log x)

So

\left.\frac{dy}{dx}\right|_{x=1}\;\;=\;\frac{-2}{1+(1)^{2}}\cdot 1^{1}(1+\log 1)

\;\;=-\;\frac{2}{2}(1)

\;\;=-1

© examsnet.com

Go to Question:

Prev Question

Next Question