Differentiate the following functions with respect to x: ^ -1 1-x… - Vidyadip

Question

Differentiate the following functions with respect to x:
$\sin^{-1}\Big\{\sqrt{\frac{1-\text{x}}{2}}\Big\},0<\text{x}<1$

✓

Answer

Let $\text{y}=\sin^{-1}\Big\{\sqrt{\frac{1-\text{x}}{2}}\Big\}$
Put $\text{x}=\cos2\theta$
$\text{y}=\sin^{-1}\Big\{\sqrt{\frac{1+\cos2\theta}{2}}\Big\}$
$=\sin^{-1}\Big\{\sqrt{\frac{2\sin^2\theta}{2}}\Big\}$
$\text{y}=\sin^{-1}(\sin\theta)\ .....(\text{i})$
Here, $0<\text{x}<1$
$\Rightarrow\ 0<\cos2\theta<1$
$\Rightarrow\ 0<2\theta<\frac{\pi}{2}$
$\Rightarrow\ 0<\theta<\frac{\pi}{4}$
So, from equation (i),
$\text{y}=\theta$
$\Big[\text{Since, } \sin^{-1}(\sin\theta)=\theta\text{ if }\theta \in\Big[-\frac{\pi}{2},\frac{\pi}{2}\Big]\Big]$
$\text{y}=\frac{1}{2}\cos^{-1}\text{x}\ \big[\text{Since x}=\cos2\theta\big]$
Differentiating it with respect to x,
$\frac{\text{dy}}{\text{dx}}=-\frac{1}{2\sqrt{1-\text{x}^2}}$

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

Similar questions

If $\text{y}=\sin^{-1}\big(6\text{x}\sqrt{1-9\text{x}^2}\big), -\frac{1}{3\sqrt{2}}<\text{x}<\frac{1}{3\sqrt{2}},$ then find $\frac{\text{dy}}{\text{dx}}.$

Using definite intergeals, find the area of the circle $x^2+ y^2 = a^2$.

Find the points of local maxima$,$ local minima and the points of inflection of the function $f(x) = x^5 - 5x^4 + 5x^3 - 1$. Also find the corresponding local maximum and local minimum values.

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

A cooperative society of farmers has 50 hectares of land to grow two crops X and Y. The profits from crops X and Y per hectare are estimated as Rs. 10,500 and Rs. 9,000 respectively. To control weeds, a liquid herbicide has to be used for crops X and Y at the rate of 20 litres and 10 litres per hectare, respectively. Further not more than 800 litres of herbicide should be used in order to protect fish and wildlife using a pond which collects drainage from this land. How much land should be allocated to each crop so as to maximise the total profit of the society?

$\int\limits^{\frac{\pi}{4}}_{-\frac{\pi}{4}}\log(\sin\text{x}+\cos\text{x})\text{dx}$

Form the differential equation of all circles which pass through origin and whose centres lie on Y-axis.

find the area of the circle $x^2 + y^2 = 16$ which is exterior to the parabola $y^2 = 6x.$

In a hospital, there are 20 kidney dialysis machines and that the chance of any one of them to be out of service during a day is 0.02. Determine the probability that exactly 3 machines will be out of service on the same day.

Sketch the graph y = |x - 5|. Evaluate $\int\limits_{0}^{1}|\text{x}-5|\text{dx} $ . What does this value of the integral represent on the graph.