Find dy dx in the following: y= ^ -1 ( 2x 1+x^ 2 ) - Vidyadip

Question

Find $\frac{\text{dy}}{\text{ dx}} $in the following:
$\text{y}=\sin^{-1}\Bigg(\frac{2\text{x}}{1+\text{x}^{2}}\Bigg)$

✓

Answer

The given relationship is $\text{y}=\sin^{-1}\Big(\frac{2\text{x}}{1+\text{x}^{2}}\Big)$
$\text{y}=\sin^{-1}\Big(\frac{2\text{x}}{1+\text{x}^{2}}\Big)$
$\Rightarrow \sin\text{y}=\frac{2\text{x}}{1+\text{x}^{2}}$
Differenting this relationship with respect to x, we obtain
$\frac{\text{d}}{\text{dx}}(\sin\text{y})=\frac{\text{d}}{\text{dx}}\Big(\frac{2\text{x}}{1+\text{x}^{2}}\Big)$
$\Rightarrow\cos\text{y}\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\Big(\frac{2\text{x}}{1+\text{x}^{2}}\Big) ...(\text{i})$
The function$\frac{2\text{x}}{1+\text{x}^{2}}$, is of the from of $\frac{\text{u}}{\text{v}}.$
Therefore, by quotient rule, we obtain
$\frac{\text{d}}{\text{dx}}\Big(\frac{2\text{x}}{1+\text{x}^{2}}\Big)=\frac{(1+\text{x}^{2}).\frac{\text{d}}{\text{dx}}(2\text{x})-2\text{x}.\frac{\text{d}}{\text{dx}}(1 +\text{x}^{2})}{(1 +\text{x}^{2})^2}$
$=\frac{(1+\text{x}^{2}).2-2\text{x}.[0+2\text{x}]}{(1 +\text{x}^{2})^2}$$=\frac{2 + 2\text{x}^{2}-4\text{x}^{2}}{(1 +\text{x}^{2})^2}=\frac{2(1-\text{x}^{2})}{(1+\text{x}^{2})^2} ...(\text{ii})$
Also, $ \sin\text{y}=\frac{2\text{x}}{1+\text{x}^{2}}$
$\Rightarrow\cos\text{y}=\sqrt{1-\sin^{2}\text{y}}=\sqrt{1-\Big(\frac{2\text{x}}{1+\text{x}^{2}}\Big)^2}=\sqrt{\frac{(1+ \text{x}^{2})^2-4\text{x}^{2}}{(1 +\text{x}^{2})^{2}}}$
$=\sqrt{\frac{1+\text{x}^4+2\text{x}^2-4\text{x}^2}{(1+\text{x}^2)^2}}=\sqrt{\frac{(1-\text{x}^2)^2}{(1+\text{x}^2)^2}}$
$=\sqrt{\frac{(1- \text{x}^{2})^{2}}{(1 +\text{x}^{2})^{2}}}=\frac{1-\text{x}^{2}}{1+\text{x}^{2}} ...\text{(iii)}$
Form(1), (2), and (3), we obtain
$\frac{1-\text{x}^{2}}{1+\text{x}^{2}}\times\frac{\text{dy}}{\text{dx}}=\frac{2(1-\text{x}^{2})}{(1 +\text{x}^{2})^{2}}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{2}{1+\text{x}^{2}}$

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

Similar questions

If $\text{A} = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{bmatrix} \text{and B = } \text{A} = \begin{bmatrix} 2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5 \end{bmatrix}$ are two square matrices, find AB and hence solve the system of linear
equations x – y = 3, 2x + 3y + 4z = 17 and y + 2z = 7.

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.
$f(x) = 2x - x^2$ on $[0, 1]$

Solve the following systems of linear equations by cramer's rule:

$3x + y + z = 2,$

$2x - 4y + 3z = -1,$

$4x + y - 3z = -11$

A man 2 metres high walks at a uniform speed of 6km/h away from a lamp-post 6 metres high. Find the rate at which the length of his shadow increases.

Solve the system of linear equation, using matrix method 2x + y + z = 1; $x - 2y - z = \frac{3}{2};\,\,3y - 5z = 9$

Show that the vectors
$\vec{\text{a}}=\frac{1}{7}(2\hat{\text{i}}+3\hat{\text{j}}+6\hat{\text{k}}),\vec{\text{b}}=\frac{1}{7}(3\hat{\text{i}}-6\hat{\text{j}}+2\hat{\text{k}}),\vec{\text{c}}=\frac{1}{7}(6\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}})$ are mutually perpendicular unit vectors.

Find the maximum and minimum values of the function $\text{f}(\text{x})=\frac{4}{\text{x}+2}+\text{x}$

A manufacturer has three machines installed in his factory. machines I and II are capable of being operated for at most 12 hours whereas Machine III must operate at least for 5 hours a day. He produces only two items, each requiring the use of three machines. The number of hours required for producing one unit each of the items on the three machines is given in the following table:

Item	Number of hours required by the machine
	I	II	III
A	1	2	1
B	2	1	$\frac{5}{4}$

He makes a profit of Rs. 6.00 on item A and Rs. 4.00 on item B. Assuming that he can sell all that he produces, how many of each item should he produces so as to maximize his profit? Determine his maximum profit. Formulate this LPP mathematically and then solve it.

Evaluate the following intregals:
$\int\frac{\text{x}^2+1}{(2\text{x}+1)(\text{x}^2-1)}\text{ dx}$

Show that the following system of linear equations is consistent and also find solutions:
$5x +3y + 7z = 4$
$3x + 26y + 2z = 9$
$7x + 2y + 10z = 5$