Download App

CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY3 Marks
Question
Find $\frac{\text{dy}}{\text{dx}},\ \text{if y}=\sin^{-1}\text{x}+\sin^{-1} \sqrt{1-\text{x}^2},\ -1\leq\text{x}\leq1$
✓

Answer

It is given that, $\text{y}=\sin^{-1}\text{x}+\sin^{-1}\sqrt{1-\text{x}^2}$
$\therefore\ \frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\Big[\sin^{-1}\text{x}+\sin^{-1}\sqrt{1-\text{x}^2}\Big]$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}(\sin^{-1}\text{x})+\frac{\text{d}}{\text{dx}}(\sin^{-1}\sqrt{1-\text{x}^2})$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{1}{\sqrt{1-\text{x}^2}}+\frac{1}{\sqrt{1-(\sqrt{1-\text{x}^2})^2}}.\frac{\text{d}}{\text{dx}}(\sqrt{1-\text{x}^2})$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{1}{\sqrt{1-\text{x}^2}}+\frac{1}{\text{x}}.\frac{1}{2\sqrt{1-\text{x}^2}}.\frac{\text{d}}{\text{dx}}(1-\text{x}^2)$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{1}{\sqrt{1-\text{x}^2}}+\frac{1}{2\text{x}\sqrt{1-\text{x}^2}}(-2\text{x})$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{1}{\sqrt{1-\text{x}^2}}-\frac{1}{\sqrt{1-\text{x}^2}}$
$\therefore\ \frac{\text{dy}}{\text{dx}}=0$

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 "3 Marks Question" 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 shortest distance between the lines whose vector equations are $\vec r = \hat i + 2\hat j + 3\hat k$ + $\lambda \left( {\hat i - 3\hat j + 2\hat k} \right)$ and $\vec r = 4\hat i + 5\hat j + 6\hat k$ + $\mu \left( {2\hat i + 3\hat j + \hat k} \right)$
→
Solve the following linear programming problem graphically:
Maximise Z = 4x + y subject to the constraints:
x + y $\le$ 50
3x + y $\le$ 90
x $\ge$ 0, y $\ge$ 0
→
If the mean and variance of a binomial variate X are 2 and 1 respectively, find P (X > 1).
→
Find the points on the curve $y = x^3 $ at which the slope of the tangent is equal to the $y-$coordinate of the point.
→
If the matric $\text{A}=\begin{bmatrix}5 & 2&\text{x} \\\text{y} & \text{z}&-3\\4&\text{t}&-7\end{bmatrix}$ is a symmetric matrix, find $x, y, z$ and $t$.
→
Evaluate the following integrals:
$\int\Big(\frac{\text{x}+1}{\text{x}}\Big)(\text{x}+\log\text{x})^2\text{dx}$
→
Using determinants, find the value of k so that the points $(k, 2 - 2k), (-k + 1, 2k)$ and $(-4 - k, 6 - 2k)$ may be collinear.
→
Solve system of linear equations, using matrix method.
x - y + 2z=7
3x + 4y - 5z = -5
2x - y + 3z = 12
→
Evaluate the definite integral in Exercise:
$\int\limits_{0}^{\frac{\pi}{2}}\cos^{2}\text{x}\ \text{dx}$
→
Integrate the rational function in exercise:
$\frac{\cos\text{x}}{(1-\sin\text{x})(2-\sin\text{x})}$
[Hint: Put sin x = t]
→