Download App

CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
Differentiate the following functions with respect to x:
$\log\sqrt{\frac{1-\cos\text{x}}{1+\cos\text{x}}}$
✓

Answer

Let, $\text{y}=\log\sqrt{\frac{1-\cos\text{x}}{1+\cos\text{x}}}$
$\Rightarrow\ \text{y}=\log\Big(\frac{1-\cos\text{x}}{1+\cos\text{x}}\Big)^\frac{1}{2}$
$\Rightarrow\ \text{y}=\frac{1}{2}\log\Big(\frac{1-\cos\text{x}}{1+\cos\text{x}}\Big) \big[\text{Using }\log\text{a}^\text{b}=\text{b}\log\text{a}\big]$
Differentiate with respect to x we get,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\Big\{\frac{1}{2}\log\Big(\frac{1-\cos\text{x}}{1+\cos\text{x}}\Big)\Big\}$
$=\frac{1}{2}\times\frac{1}{\Big(\frac{1-\cos\text{x}}{1+\cos\text{x}}\Big)}\times\frac{\text{d}}{\text{dx}}\Big(\frac{1-\cos\text{x}}{1+\cos\text{x}}\Big)$
[Using chain rule]
$=\frac{1}{2}\Big(\frac{1+\cos\text{x}}{1-\cos\text{x}}\Big)\bigg[\frac{(1+\cos\text{x})\frac{\text{d}}{\text{dx}}(1-\cos\text{x})-(1-\cos\text{x})\frac{\text{d}}{\text{dx}}(1+\cos\text{x})}{(1+\cos\text{x})^2}\bigg]$
$=\frac{1}{2}\Big(\frac{1+\cos\text{x}}{1-\cos\text{x}}\Big)\bigg[\frac{(1+\cos\text{x})(\sin\text{x})-(1-\cos\text{x})(-\sin\text{x})}{(1+\cos\text{x})^2}\bigg]$
$=\frac{1}{2}\Big(\frac{1+\cos\text{x}}{1-\cos\text{x}}\Big)\Big[\frac{\sin\text{x}+\sin\text{x}\cos\text{x}+\sin\text{x}-\sin\text{x}\cos\text{x}}{(1+\cos\text{x})^2}\Big]$
$=\frac{1}{2}\Big(\frac{1+\cos\text{x}}{1-\cos\text{x}}\Big)\Big[\frac{2\sin\text{x}}{(1+\cos\text{x})^2}\Big]$
$=\frac{\sin\text{x}}{(1-\cos\text{x})(1+\cos\text{x})}$
$=\frac{\sin\text{x}}{1-\cos^2\text{x}}$
$=\frac{\sin\text{x}}{\sin^2\text{x}}$
$=\frac{1}{\sin\text{x}}$
$=\text{cosec x}$
So,
$\frac{\text{d}}{\text{dx}}\Big(\log\sqrt{\frac{1-\cos\text{x}}{1+\cos\text{x}}}\Big)=\text{cosec x}$

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→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the equation of a curve passing through the point (0, 1). If the slope of the tangent to the curve at any point (x, y) is equal to the sum of the x coordinate (abscissa) and the product of the x coordinate and y coordinate (ordinate) of that point.
→
Write the set of values of 'a' for which $\text{f}(\text{x})=\log_\text{a}\text{x}$ is decreasing in its domain.
→
Differentiate $\sin^{-1}\Big(2\text{ax}\sqrt{1-\text{a}^2\text{x}^2}\Big)$ with respect to $\sqrt{1-\text{a}^2\text{x}^2},$ if $-\frac{1}{\sqrt{2}}<\text{ax}<\frac{1}{\sqrt{2}}$.
→
Find the inverse of the following matrices:$\begin{bmatrix}2 & -1 & 1 \\ -1 & 2 & -1\\ 1 & -1 & 2 \end{bmatrix}$
→
Find the distance between the lines $l_1$ and $l_2$ given by
$\vec r = \hat i + 2\hat j - 4\hat k + \lambda (2\hat i + 3\hat j + 6\hat k)$
and $\vec r = 3\hat i + 3\hat j - 5\hat k + \mu (2\hat i + 3\hat j + 6\hat k)$
→
Find the equation of the normal lines to the curve $3x^2 - y^2 = 8$ which are parallel to the line $x + 3y = 4$.
→
For each of the differential equations in find a particular solution satisfying the given condition:
$(\text{x}^3+\text{x}^2+\text{x}+1) \frac{\text{dy}}{\text{dx}} = 2\text{x}^2+\text{x; y} =1 \ \text{when x = 0}$
→
$\int\limits_{0}^{\pi}\text{x}\log\sin\text{x dx}$
→
Find the absolute maximum and the absolute minimum value of the following functions in the given intervals:
$f(x) = 3x^4 - 8x^3 + 12x^2 - 48x + 25$ in $[0,3]$
→
Using integration find the are of the region bounded by the parabola $y^{2} ={4x}$ and the circle $4x^{2} + 4y^{2} = 9$
→