Download App

Differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferentiation5 Marks
Question
If $\text{y}=\frac{\text{x}\sin^{-1}\text{x}}{\sqrt{1-\text{x}^2}},$ prove that $(1-\text{x}^2)\frac{\text{dy}}{\text{dx}}=\text{x}+\frac{\text{y}}{\text{x}}$
✓

Answer

Givne, $\text{y}=\frac{\text{x}\sin^{-1}\text{x}}{\sqrt{1-\text{x}^2}}$
Differentiate with respect to x,
$\frac{\text{d}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\Big(\frac{\text{x}\sin^{-1}\text{x}}{\sqrt{1-\text{x}^2}}\Big)$
$=\bigg[\frac{\sqrt{1-\text{x}^2}\frac{\text{d}}{\text{dx}}(\text{x}\sin^{-1}\text{x})-(\text{x}\sin^{-1}\text{x})\frac{\text{d}}{\text{dx}}(\sqrt{1-\text{x}^2})}{(\sqrt{1-\text{x}^2})^2}\bigg]$
[Using quotient rule, product rule, chain rule]
$=\begin{bmatrix}\frac{\sqrt{1-\text{x}^2}\Big\{\text{x}\frac{\text{d}}{\text{dx}}\sin^{-1}\text{x}+\sin^{-1}\text{x}\frac{\text{d}}{\text{dx}}(\text{x})\Big\}-\big(\text{x}\sin^{-1}\text{x}\big)\frac{1}{2\sqrt{1-\text{x}^2}}\frac{\text{d}}{\text{dx}}\big(1-\text{x}^2\big)}{\big(1-\text{x}^2\big)} \end{bmatrix}$
$=\begin{bmatrix}\frac{\sqrt{1-\text{x}^2}\Big\{\frac{\text{x}}{\sqrt{1-\text{x}^2}}+\sin^{-1}\text{x}\Big\}-\frac{\text{x}\sin{-1}\text{x}(-2\text{x})}{2\sqrt{1-\text{x}^2}}}{\big(1-\text{x}^2\big)} \end{bmatrix}$
$=\begin{bmatrix}\frac{\text{x}+\sqrt{1-\text{x}^2}\sin^{-1}\text{x}+\frac{\text{x}^2\sin^{-1}\text{x}}{\sqrt{1-\text{x}^2}}}{\big(1-\text{x}^2\big)} \end{bmatrix}$
$\Rightarrow\big(1-\text{x}^2\big)\frac{\text{dy}}{\text{dx}}=\text{x}+\frac{\sqrt{1-\text{x}^2}\sin^{-1}}{1}+\frac{\text{x}^2\sin^{-1}\text{x}}{\sqrt{1-\text{x}^2}}$
$\Rightarrow\big(1-\text{x}^2\big)\frac{\text{dy}}{\text{dx}}=\text{x}+\bigg(\frac{(1-\text{x}^2)\sin^{-1}\text{x}+\text{x}^2\sin^{-1}\text{x}}{\sqrt{1-\text{x}^2}}\bigg)$
$\Rightarrow\big(1-\text{x}^2\big)\frac{\text{dy}}{\text{dx}}=\text{x}+\bigg(\frac{\sin^{-1}\text{x}-\text{x}^2\sin^{-1}\text{x}+\text{x}^2\sin^{-1}\text{x}}{\sqrt{1-\text{x}^2}}\bigg)$
$\Rightarrow\big(1-\text{x}^2\big)\frac{\text{dy}}{\text{dx}}=\text{x}+\bigg(\frac{\sin^{-1}\text{x}}{\sqrt{1-\text{x}^2}}\bigg)$
$\Rightarrow(1-\text{x}^2)\frac{\text{dy}}{\text{dx}}=\text{x}+\frac{\text{y}}{\text{x}}\ \Big\{\text{Since, given y}=\frac{\text{x}\sin^{-1}\text{x}}{\sqrt{1-\text{x}^2}}\Big\}$

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

Integrate the function $\frac{e^{a \tan ^{-1} x}}{\left(1+x^2\right)^{3 / 2}}$ with respect to $x$.
→
If $\text{y}=\text{x}^\text{n}\{\text{a}\cos(\log\text{x})+\text{b}\sin(\log\text{x})\},$ prove that $\text{x}^2\frac{\text{d}^2\text{y}}{\text{dx}^2}+(1-2\text{n})\frac{\text{dy}}{\text{dx}}+(1+\text{n}^2)\text{y}=0.$
→
Consider $f :\{1, 2, 3\} \rightarrow \{a, b, c\}$ and $g : \{a, b, c\} \rightarrow \{$apple, ball, cat$\} $defined as $f(1) = a, f(2) = b, f(3) = c, g(a) =$ apple, $g(b) =$ ball and $g(c) =$ cat. Show that $f, g$ and gof are invertible. Find $f^{-1}, g^{-1}$ and $gof^{-1}$ and show that $(gof)^{-1} = f^{-1}og^{-1}.$
→
Prove that:
$\begin{vmatrix}\text{a}^2&\text{bc}&\text{ac}+\text{c}^2\\\text{a}^2+\text{ab}&\text{b}^2&\text{ac}\\\text{ab}&\text{b}^2+\text{ac}&\text{c}^2\end{vmatrix}=4\text{a}^2\text{b}^2\text{c}^2$
→
An urn contains 4 red and 7 black balls. Two balls are drawn at random with replacement. Find the probability of getting,
  1. 2 red balls,
  2. 2 blue balls,
  3. One red and one blue ball.
→
If $\text{y}\log(1+\cos\text{x}),$ prove that $\frac{\text{d}^3\text{y}}{\text{dx}^3}+\frac{\text{d}^\text{y}}{\text{dx}^2}.\frac{\text{d}\text{y}}{\text{dx}}=0$
→
Using differentials, find the approximate values of the following:
$(1.999)^5$
→
Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{4\text{x}}{1-4\text{x}^2}\Big),-\frac{1}{2}<\text{x}<\frac{1}{2}$
→
If $\text{y}=1+\frac{\alpha}{\big(\frac{1}{\text{x}}-\alpha\big)}+\frac{\frac{\beta}{\text{x}}}{\big(\frac{1}{\text{x}}-\alpha\big)\big(\frac{1}{\text{x}}-\beta\big)}+\frac{\frac{\gamma}{\text{x}^2}}{\big(\frac{1}{\text{x}}-\alpha\big)\big(\frac{1}{\text{x}}-\beta\big)\big(\frac{1}{\text{x}}-\gamma\big)},$ find $\frac{\text{dy}}{\text{dx}}$
→
If $\text{y}=(\sin\text{x})^{(\sin\text{x})^{(\sin\text{x})^{....\infty}}},$ prove that $\frac{\text{y}^2\cot\text{x}}{(1-\text{y}\log\sin\text{x})}$
→