Download App

Continuity and Differentiability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity and Differentiability4 Marks
Question
Differentiate the function (sin x)x + sin–1$\sqrt{x}$ w.r.t. x.
✓

Answer

Given function is: $(\sin x)^{x}+\sin ^{-1} \sqrt{x}$ 
Let $y=(\sin x)^{x}+\sin ^{-1} \sqrt{x}$ 
Let y = u + v
$\Rightarrow$ u = (sin x)x and $\mathrm{v}=\sin ^{-1} \sqrt{\mathrm{x}}$ 
For, $\mathrm{u}=(\sin \mathrm{x})^{\mathrm{x}}$ 
Taking log on both sides, we get
log u = log(sin x)x 
$\Rightarrow$ log u = x.log sin x
Now, differentiate both sides with respect to x
$\frac{d}{d x}(\log u)=\frac{d}{d x}[x \cdot \log (\sin x)]$ 
$\left.\Rightarrow \frac{1}{\mathrm{u}} \frac{\mathrm{du}}{\mathrm{dx}}=\mathrm{x} \cdot \frac{\mathrm{d}}{\mathrm{dx}} \log (\sin \mathrm{x})\right)+\log (\sin \mathrm{x}) \cdot \frac{\mathrm{d}}{\mathrm{dx}}(\mathrm{x})$ 
$\Rightarrow \frac{\mathrm{du}}{\mathrm{dx}}=\mathrm{u}\left[\mathrm{x} \cdot \frac{1}{\sin \mathrm{x}} \frac{\mathrm{d}}{\mathrm{dx}}(\sin \mathrm{x})+\log (\sin \mathrm{x}) \cdot(1)\right]$ 
$\Rightarrow \frac{\mathrm{du}}{\mathrm{dx}}=(\sin \mathrm{x})^{\mathrm{x}}\left[\frac{\mathrm{x}}{\sin \mathrm{x}} \cdot \cos \mathrm{x}+\log (\sin \mathrm{x}) \cdot(1)\right]$ 
$\Rightarrow \frac{d u}{d x}=(\sin x)^{x}[x \cdot \cot x+\log \sin x]$ 
For, v = $\sin ^{-1} \sqrt{x}$ 
Now, differentiating both sides with respect x
$\frac{d v}{d x}=\frac{d}{d x}\left[\sin ^{-1} \sqrt{x}\right]$ 
$\Rightarrow \frac{d v}{d x}=\frac{1}{\sqrt{1-(\sqrt{x})^{2}}} \cdot \frac{d}{d x}(\sqrt{x})$ 
$\Rightarrow \frac{d v}{d x}=\frac{1}{\sqrt{1-x}} \cdot \frac{1}{2(\sqrt{x})}$ 
$\Rightarrow \frac{d v}{d x}=\frac{1}{2 \sqrt{x-x^{2}}}$ 
Because, y = u + v
$\Rightarrow \frac{d y}{d x}=\frac{d u}{d x}+\frac{d v}{d x}$ 
$\Rightarrow \frac{d y}{d x}=(\sin x)^{x}[x \cdot \cot x+\log \sin x]+\frac{1}{2 \sqrt{x-x^{2}}}$

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 "4 Marks" 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

Show that the function f : Q → Q, defined by f(x) = 3x + 5, is invertible. Also, find f-1.
→
Evaluate the following integrals:
$\int\text{x}\cos^3\text{x dx}$
→
Evaluate the following integrals:
$\int^\limits{\frac{\pi}{2}}_\frac{\pi}{3}\frac{\sqrt{1+\cos\text{x}}}{(1-\cos\text{x})^{\frac{3}{2}}}\text{ dx}$
→
Solve the following differential equation
$(1+\text{x}^2)\text{dy}=\text{xy dx}$
→
Find the area of the smaller part of the circle x2 + y2 = a2 cut off by the line $\text{x}=\frac{\text{a}}{\sqrt2}.$
→
Find the area of the region bounded by $\text{y}=\sqrt{\text{x}}$ and y = x.
→
Express the matrix $\text{A}=\begin{bmatrix}4&2&-1 \\3 & 5&7\\1&-2&1 \end{bmatrix}$ as the sum of a symmetric and a skew-symmetric matrix.
→
Find the shortest distance between the lines whose vector equations are:
$\vec{\text{r}}=\Big(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}\Big)+\lambda\Big(\hat{\text{i}}-3\hat{\text{j}}+2\hat{\text{k}}\Big)$
$\text{and}\ \vec{\text{r}}=4\hat{\text{i}}+5\hat{\text{j}}+6\hat{\text{k}}+\mu\Big(2\hat{\text{i}}+3\hat{\text{j}}+\hat{\text{k}}\Big)$
→
Evaluate the following:
$\begin{vmatrix}0&\text{xy}^2&\text{xz}^2\\\text{x}^2\text{y}&0&\text{yz}^2\\\text{x}^2\text{z}&\text{zy}^2&0\end{vmatrix}$
→
Find the points o local maxima or local minima, if any, of the following functions, using the first derivatives test. Also, find the local maximum or local minimum values, as the case may be:

f(x) = x3(x - 1)2

→