Download App

Differentiation — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDifferentiation5 Marks
Question
Differentiate the following functions with respect to x:
$\cos^{-1}\Big(\frac{\text{x}+\sqrt{1-\text{x}^2}}{\sqrt{2}}\Big),-1<\text{x}<1$
✓

Answer

Let $\text{y}=\cos^{-1}\Big(\frac{\text{x}+\sqrt{1-\text{x}^2}}{\sqrt{2}}\Big)$
Put $\text{x}=\cos\theta$
$\text{y}=\sin^{-1}\Big\{\frac{\cos\theta+\sqrt{1-\cos^2\theta}}{\sqrt{2}}\Big\}$
$\text{y}=\cos^{-1}\Big\{\frac{\cos\theta+\sin\theta}{\sqrt{2}}\Big\}$
$\text{y}=\cos^{-1}\Big\{\cos\theta\Big(\frac{1}{\sqrt{2}}\Big)+\sin\theta\Big(\frac{1}{\sqrt{2}}\Big)\Big\}$
$\text{y}=\cos^{-1}\Big\{\cos\theta\cos\frac{\pi}{2}+\sin\theta\sin\frac{\pi}{2}\Big\}$
$\text{y}=\cos^{-1}\Big\{\cos\Big(\theta-\frac{\pi}{4}\Big)\Big\}\ .....(\text{i})$
Here, $-1<\text{x}<1$
$\Rightarrow -1<\cos\theta<1$
$\Rightarrow\frac{3\pi}{4}<\theta<\frac{5\pi}{4}$
$\Rightarrow\Big(\frac{3\pi}{4}-\frac{\pi}{4}\Big)<\Big(\theta-\frac{\pi}{4}\Big)<\frac{5\pi}{4}-\frac{\pi}{4}$
$\Rightarrow\Big(\frac{\pi}{4}\Big)<\Big(\theta-\frac{\pi}{4}\Big)<\pi$
So, from equation (i),
$\text{y}=\Big(\theta-\frac{\pi}{4}\Big)\ \big[\text{Since}, \cos^{-1}(\cos\theta)=\theta,\text{ if }\theta\in[0,\pi]\big]$
$\text{y}=\cos^{-1}\text{x}-\frac{\pi}{4}\big[\text{Since, x}=\sin\theta\big]$
Differentiating it with respect to x,
$\frac{\text{dy}}{\text{dx}}=-\frac{1}{\sqrt{1-\text{x}^2}}+0$
$\frac{\text{dy}}{\text{dx}}=-\frac{1}{\sqrt{1-\text{x}^2}}$

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 "Solve the Following Question.(5 Marks)" from Differentiation→Differentiation — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

$\int\text{x}\sqrt{\text{x}+2}\ \text{dx}$
→
Show that the following system of linear equation is inconsistent:
$2x + 5y = 7$
$6x + 15y = 13$
→
Find gof and fog when $f: R \rightarrow R$ and $g: R \rightarrow R$ are defined by: $f(x)=2 x+3$ and $g(x)=x^2+5$
→
Evaluate the following integrals:
$\int\frac{\text{x}^2-1}{\text{x}^4+1}\ \text{dx}$
→
Verify Rolle's theorem for the following function on the indicated intervals$f(x) = x(x^- 4)^2$ on the interval $[0, 4]$
→
Solve the following systems of linear equations by cramer's rule:
2x - 3y - 4z = 29,
-2x + 5y - z = -15,
3x - y + 5z = -11
→
Evalute the following integrals:
$\int\frac{1-\cot\text{x}}{1+\cot\text{x}}\text{dx}$
→
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.
$f(x) = x^2- 1$ on $[2, 3]$
→
On Z, the set of all integers, a binary operation * is defined by a * b = a + 3b - 4. Prove that * is neither commutative nor associative on Z.
→
Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12cm is 16cm.
→