Download App

Differentiation — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDifferentiation5 Marks
Question
Differentiate the following functions with respect to x:
$\sin^{-1}\Big(\frac{\text{x}+\sqrt{1-\text{x}^3}}{\sqrt{2}}\Big),-1<\text{x}<1$
✓

Answer

Let $\text{y}=\sin^{-1}\Big(\frac{\text{x}+\sqrt{1-\text{x}^3}}{\sqrt{2}}\Big)$
Put $\text{x}=\text{a}\sin\theta,\text{ So}$
$=\sin^{-1}\Big\{\frac{\sin\theta+\sqrt{1-\sin^2\theta}}{\sqrt{2}}\Big\}$
$=\sin^{-1}\Big\{\frac{\sin\theta+\cos\theta}{\sqrt{2}}\Big\}$
$=\sin^{-1}\Big\{\sin\theta\Big(\frac{1}{\sqrt{2}}\Big)+\cos\theta\Big(\frac{1}{\sqrt{2}}\Big)\Big\}$
$=\sin^{-1}\Big\{\sin\theta\cos\frac{\pi}{4}+\cos\theta\sin\frac{\pi}{4}\Big\}$
$\text{y}=\sin^{-1}\Big\{\sin\Big(\theta+\frac{\pi}{4}\Big)\Big\}\ .....(\text{i})$
Here, $-1<\text{x}<1$
$\Rightarrow\ -1<\sin\theta<1$
$\Rightarrow -\frac{\pi}{2}<\theta<\frac{\pi}{2}$
$\Rightarrow\Big(-\frac{\pi}{2}+\frac{\pi}{4}\Big)<\Big(\frac{\pi}{4}+\theta\Big)<\frac{3\pi}{4}$
So, from equation (i),
$\text{y}=\theta+\frac{\pi}{4}\Big[\text{Since},\sin^{-1}(\sin\theta)=\theta,\text{ as }\theta\in\Big[-\frac{\pi}{2},\frac{\pi}{2}\Big]\Big]$
$\text{y}=\sin^{-1}\text{x}+\frac{\pi}{4} \big[\text{Since},\sin\theta=\text{x}\big]$
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

Evaluate the following intergrals:
$\int\text{e}^\text{ax}\sin(\text{bx}+\text{c})\text{dx}$
→
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.
$f(x) = x^3- 5x^2 - 3x$ on $[1, 3]$
→
Show that the function $f : Q \rightarrow Q$, defined by $f(x) = 3x + 5$, is invertible. Also, find $f^{-1}$​​​​​​​.
→
Evaluate the following integrals:
$\int\frac{\text{x}}{(\text{x}^2+2\text{x}+2)\sqrt{\text{x}+1}}\text{ dx}$
→
Evaluate the following integrals:$\int\limits^{\frac{\pi}{4}}_{-\frac{\pi}{4}}\sin^2\text{x dx}$
→
Evaluate the following integrals:
$\int\tan\text{x}\sec^2\text{x}\sqrt{1-\tan^2\text{x}}\text{ dx}$
→
Find all vectors of magnitude $10\sqrt{3}$ that are perpendicular to the plane of $\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}}$ and $-\hat{\text{i}}+3\hat{\text{i}}+4\hat{\text{k}}.$
→
Find the feasible solution of the following inequations graphically.3x + 4y ≥ 12, 4x + 7y ≤ 28, y ≥ 1, x ≥ 0
→
If $\text{P}=\begin{bmatrix}\text{x}&0&0\\0&\text{y}&0\\0&0&\text{z}\end{bmatrix}$ and $\text{Q}=\begin{bmatrix}\text{a}&0&0\\0&\text{b}&0\\0&0&\text{c}\end{bmatrix},$ prove that $\text{PQ}=\begin{bmatrix}\text{xa}&0&0\\0&\text{y}\text{b}&0\\0&0&\text{zc}\end{bmatrix}=\text{QP}$
→
Evaluate the following integrals:$\int\limits^0_{-5}\text{f(x)}\text{dx,}$ Where $\text{f(x)}=|\text{x}|+|\text{x}+2|+|\text{x}+5|$
→