Find dy dx ,if y = sin^ -1 [x1 - x-x 1 - x^ 2 ]. - Vidyadip

Question

Find $\frac{\text{dy}}{\text{dx}},\text{if y = sin}^{-1}[\text{x}\sqrt{1 - x}-\sqrt{x}\sqrt{1 - x^{2}}]$.

✓

Answer

$\text{y}=\sin^{-1}\text[{x}\sqrt{1-x}-\sqrt{x}\sqrt{1 - x^{2}}]............(i)$
Let $x = \sin\alpha\text{ and }\sqrt{x}=\sin\theta$
$\therefore$ (i) Becomes $y = \sin^{-1}[\sin\alpha\cos\theta-\cos\alpha\sin\theta]$.
$=\sin^{-1}[\sin(\alpha-\theta)]=\alpha-\theta$
$=\sin^{-1}\text{x}-\sin^{-1}\sqrt{x}$
$\therefore\frac{\text{dy}}{\text{dx}}=\frac{1}{\sqrt{1}-x^2}-\frac{1}{2\sqrt{x}\sqrt{1-x}}$

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 DIFFERENTIAL EQUATIONS→DIFFERENTIAL EQUATIONS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Three vectors $\vec{a}, \vec{b}$ and $\vec c$ satisfy the condition $\vec{a}+\vec{b}+\vec{c}=\vec{0}$. Evaluate the quantity $\mu=\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}$, if $|\vec{a}|=3,|\vec{b}|=4$ and $|\vec{c}|=2$.

Find the Vector and Cartesian equations of the line passing through the point (1, 2, – 4) and perpendicular to the two lines $\frac{\text{x - 8}}{3} =\frac{\text{y + 19}}{-16}=\frac{\text{z - 10}}{7}\text{ and }\frac{\text{x - 15}}{3}= \frac{\text{y - 29}}{8}=\frac{\text{z - 5}}{-5}$.

In a triangle OAB, $\angle\text{AOB}=90^\circ.$ If P and Q are points of trisection of AB, prove that $\text{OP}^2+\text{OQ}^2=\frac{5}{9}\text{AB}^2.$

A swimming pool is to be drained for cleaning. If $L$ represents the number of litres of water in the pool $t $ seconds after the pool has been plugged off to drain and $L = 200(10 - t)^2.$ How fast is the water running out at the end of $5$ seconds? What is the average rate at which the water flows out during the first $5$ seconds?

Show that the function $f: R \rightarrow\{x \in R:-1 < x<1\}$ defined by $f(x)=\frac{x}{1+|x|}, x \in R$ is one$-$one and onto function.

A plane passes through the point (1, -2, 5) and is perpendicular to the line joining the origin to the point $2\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}.$ Find the vector and cartesian forms of the equation of the plane.

Evaluate the following integrals:
$\int(2\text{x}+5)\sqrt{10-4\text{x}-3\text{x}^2}\text{dx}$

There are two types of fertilisers 'A' and 'B'. 'A' consists of 12 % nitrogen and 5 % phosphoric acid whereas 'B' consists of 4 % nitrogen and 5 % phosphoric acid. After testing the soil conditions, farmer finds that he needs at least 12 kg of nitrogen and 12 kg of phosphoric acid for his crops. If 'A' costs 10 per kg and 'B' cost 8 per kg, then graphically determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost.

Show that the height of the cylinder of maximum volume that can be inscribed a sphere of radius R is $\frac{2\text{R}}{\sqrt{3}}$ .

Differentiate the following functions with respect to x:
$\tan^{-1}\Big\{\frac{\text{x}}{\sqrt{\text{a}^2-\text{x}^2}}\Big\},-\text{a}<\text{x}<\text{a}$