Download App

CONTINUITY AND DIFFERENTIABILITY — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
Differentiate $\tan^{-1}\Big(\frac{\sqrt{1+\text{x}^2}-1}{\text{x}}\Big)$ w.r.t. $\tan^{-1}\text{x}$ when $\text{x}\neq0.$
✓

Answer

Let $\text{u}=\tan^{-1}\frac{\sqrt{1+\text{x}^2}-1}{\text{x}}$ and $\text{y}=\tan^{-1}\text{x}$
Put $\text{x}=\tan\theta$
$\Rightarrow\ \text{u}=\tan^{-1}\frac{\sqrt{1+\tan^2\theta}-1}{\tan\theta}$
$=\tan^{-1}\frac{\sec\theta-1}{\tan\theta}=\tan^{-1}\Big(\frac{1-\cos\theta}{\sin\theta}\Big)$
$=\tan^{-1}\bigg[\frac{2\sin^2\frac{\theta}{2}}{2\sin\frac{\theta}{2}\cos\frac{\theta}{2}}\bigg]$ $=\tan^{-1}\Big[\tan\frac{\theta}{2}\Big]=\frac{\theta}{2}=\frac{1}{2}\tan^{-1}\text{x}$
$\therefore\ \frac{\text{du}}{\text{dx}}=\frac{1}{2}\cdot\frac{1}{1+\text{x}^2}\ \ \dots(\text{i})$
and $\frac{\text{dv}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\tan^{-1}\text{x}=\frac{1}{1+\text{x}^2}\ \ \dots(\text{ii})$
$\therefore\ \frac{\text{du}}{\text{dv}}=\frac{\frac{\text{du}}{\text{dx}}}{\frac{\text{dv}}{\text{dx}}}=\frac{1}{2}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find the local maximum and local minima, of the function $\text{f(x)} = \sin x - \cos x, 0< x < 2\pi.$ Also find the local maximum and local minimum values.
→
Evaluate the following integrals:
$\int\limits^4_{0}\big(|\text{x}|+|\text{x}+2|+|\text{x}+4|\big)\text{dx}$ 
→
Discuss the continuity of the f(x) at the indicated points f(x) = |x| + |x - 1| at x = 0, 1.
→
Prove that
$\text{L.H.S}=\sin\Big\{\tan^{-1}\frac{1-\text{x}^2}{2\text{x}}+\cos^{-1}\frac{1-\text{x}^2}{1+\text{x}^2}\Big\}=1$
→
Using differentials, find the approximate values of the following:
$\log_\text{e}4.04,$ it being given that $\log_{{10}^{4}}=0.6021$ and $\log_{10}\text{e}=0.4343$
→
Find the absolute maximum and minimum values of a function f given by
$\text{f}(\text{x})=12\text{x}^\frac{4}{3}-6\text{x}^\frac{1}{3},\text{x}\in[-1,1]$
→
Evaluate the following intregals:
$\int\frac{1}{4\cos^2\text{x}+3\sin^2\text{x}}\ \text{dx}$
→
Solve the following differential equation
$\sin^4\text{x}\frac{\text{dy}}{\text{dx}}=\cos\text{x}$
→
Evaluate the following integrals as limit of sum:
$\int\limits^5_{0}(\text{x}+1)\text{dx}$
→
Solve the following system of homogeneous linear equations:
x + y - 2z = 0,
2x + y - 3z = 0,
5x + 4y - 9z = 0
→