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

Check the commutativity and associativity of the following binary operations:
'*' on Q defined by $a * b = (a - b)^2$ for all $a, b \in Q.$
→
For two matrices A and B, $\text{A}=\begin{bmatrix}2&1&3\\4&1&0\end{bmatrix},\text{B}=\begin{bmatrix}1&-1\\0&2\\5&0\end{bmatrix}$ verify that $(AB)^T = B^TA^T$.
→
Discuss the applicability of the Rolle's theorem for the following function on the indicated interval
$\text{f}(\text{x})=\text{x}^{\frac{2}{3}}\text{ on }[-1,1]$
→
Evaluate the following integrals:
$\int\frac{\text{cosec }\text{x}}{\text{cosec }\text{x}-\cot\text{x}}\text{dx}$
→
Find the point on the curve $y^2 = 8x$ for which the abscissa and ordinate change at the same rate.
→
When the curve crosses the $y-$axis, the point on the curve is of the form $(0, y).$
→
Find the value of x if the area of $\triangle$ is $35$ square cms with vertices $(x, 4), (2, -6)$ and $(5, 4)$.
→
If with reference to the right handed system of mutually perpendicular unit vectors $\hat i,\hat j$ and $\hat k$, $\vec \alpha = 3\hat i - \hat j$, $\vec \beta = 2\hat i + \hat j - 3\hat k$, then express $\vec \beta $ in the form $\vec \beta = {\vec \beta _1} + {\vec \beta _2}$, where ${\vec \beta _1}$ is || to $\vec \alpha $ and ${\vec \beta _2}$ is perpendicular to $\vec \alpha $.
→
Let $+_6 ($addition modulo $6)$ be a binary operation on $S = \{0, 1, 2, 3, 4, 5\}$. Write the value of $2 + _64^{−1}+ _63^{−1}.$
→
Find the value of $\tan^{-1}\Big(\frac{\text{x}}{\text{y}}\Big)-\tan^{-1}\Big(\frac{\text{x-y}}{\text{x+y}}\Big)$
→