Download App

CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
Prove that $\frac{\text{dy}}{\text{dx}}\Big\{\frac{\text{x}}{2}\sqrt{\text{a}^2-\text{x}^2}+\frac{\text{a}^2}{2}\sin^{-1}\frac{\text{x}}{\text{a}}\Big\}=\sqrt{\text{a}^2-\text{x}^2}$
✓

Answer

$\frac{\text{dy}}{\text{dx}}\Big\{\frac{\text{x}}{2}\sqrt{\text{a}^2-\text{x}^2}+\frac{\text{a}^2}{2}\sin^{-1}\frac{\text{x}}{\text{a}}\Big\}=\sqrt{\text{a}^2-\text{x}^2}$
$\text{L.H.S}=\frac{\text{dy}}{\text{dx}}\Big\{\frac{\text{x}}{2}\sqrt{\text{a}^2-\text{x}^2}+\frac{\text{a}^2}{2}\sin^{-1}\frac{\text{x}}{\text{a}}\Big\}$
$=\frac{\text{dy}}{\text{dx}}\Big(\frac{\text{x}}{2}\sqrt{\text{a}^2-\text{x}^2}\Big)+\frac{\text{d}}{\text{dx}}\Big(\frac{\text{a}^2}{2}\sin^{-1}\frac{\text{x}}{\text{a}}\Big)$
$=\frac{1}{2}\Big[\text{x}\frac{\text{d}}{\text{dx}}\sqrt{\text{a}^2-\text{x}^2}+\sqrt{\text{a}^2-\text{x}^2}\frac{\text{d}}{\text{dx}}(\text{x})\Big] \\ +\frac{\text{a}^2}{2}\times\frac{1}{\sqrt{1-\Big(\frac{\text{x}}{\text{x}}\Big)^2}}\times\frac{\text{d}}{\text{dx}}\Big(\frac{\text{x}}{\text{a}}\Big)$
[Using product rule, chain rule]
$=\frac{1}{2}\bigg[\text{x}\times\frac{1}{2\sqrt{\text{a}^2-\text{x}^2}}\frac{\text{d}}{\text{dx}}\big(\text{a}^2-\text{x}^2\big)+\sqrt{\text{a}^2-\text{x}^2}\Big] \\ +\Big(\frac{\text{a}^2}{2}\Big)\times\frac{1}{\sqrt{\frac{\text{a}^2-\text{x}^2}{\text{a}^2}}}\times\Big(\frac{1}{\text{a}}\Big)$
$=\frac{1}{2}\Big[\frac{\text{x}(-2\text{x})}{2\sqrt{\text{a}^2-\text{x}^2}}+\sqrt{\text{a}^2-\text{x}^2}\Big]+\Big(\frac{\text{a}^2}{2}\Big)\times\frac{1}{\sqrt{\text{a}^2-\text{x}^2}}\times\Big(\frac{1}{\text{a}}\Big)$
$=\frac{1}{2}\bigg[\frac{-2\text{x}^2+2\big(\text{a}^2-\text{x}^2\big)}{2\sqrt{\text{a}^2-\text{x}^2}}\bigg]+\frac{\text{a}^2}{2\sqrt{\text{a}^2-\text{x}^2}}$
$=\frac{1}{2}\bigg[\frac{2\big(\text{a}^2-2\text{x}^2\big)}{2\sqrt{\text{a}^2-\text{x}^2}}\bigg]+\frac{\text{a}^2}{2\sqrt{\text{a}^2-\text{x}^2}}$
$=\frac{\text{a}^2-2\text{x}^2}{2\sqrt{\text{a}^2-\text{x}^2}}+\frac{\text{a}^2}{2\sqrt{\text{a}^2-\text{x}^2}}$
$=\frac{\text{a}^2-2\text{x}^2+\text{a}^2}{2\sqrt{\text{a}^2-\text{x}^2}}$
$=\frac{2\text{a}^2-2\text{x}^2}{2\sqrt{\text{a}^2-\text{x}^2}}$
$=\frac{2\big(\text{a}^2-\text{a}^2\big)}{2\sqrt{\text{a}^2-\text{x}^2}}$
$=\frac{(\text{a}^2-\text{x}^2)}{\sqrt{\text{a}^2-\text{x}^2}}$
$=\sqrt{\text{a}^2-\text{x}^2}$
$=\text{R.H.S}$

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 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

If $x + y + z = 0$, then prove that $\begin{vmatrix}\text{xa}&\text{yb}&\text{zc}\\\text{yc}&\text{za}&\text{xb}\\\text{zb}&\text{xc}&\text{ya}\end{vmatrix}=\text{xyz}\begin{vmatrix}\text{a}&\text{b}&\text{c}\\\text{c}&\text{a}&\text{b}\\\text{b}&\text{c}&\text{a}\end{vmatrix},$
→
Explain if Rolle's theorem is applicable to any one of the following functions.
  1. $\text{f}(\text{x})=[\text{x}]\text{ on }\text{x}\in[5,9]$
  2. $\text{f}(\text{x})=[\text{x}]\text{ on }\text{x}\in[-2,2]$
Can you say something about the converse of Rolle's Theorem from these functions?
→
Write the minors and cofactors of element of the first column of the following matrices and hence evaluate the determinant in case:
$\text{A}=\begin{vmatrix}1&\text{a}&\text{bc}\\1&\text{b}&\text{ca}\\1&\text{c}&\text{ab} \end{vmatrix}$
→
Find the equation of the normal lines to the curve $3x^2 - y^2 = 8$ which are parallel to the line $x + 3y = 4.$
→
Find the inverse of the following matrices by using elementry row transformation:$\begin{bmatrix} 3 & 0 & -1 \\ 2 & 3 & 0 \\ 0 & 4 & 1 \end{bmatrix}$
→
Differentiate the following functions with respect to x:
$\sin^{-1}\Big(\frac{\text{x}}{\sqrt{\text{x}^2+\text{x}^2}}\Big)$
→
Evaluate the following integrals:
$\int(3\text{x}+1)\sqrt{4-3\text{x}-2\text{x}^2}\text{dx}$
→
Find the condition for the following set of curves to intersect orthogonally
$\frac{\text{x}^2}{\text{a}^2}-\frac{\text{y}^2}{\text{b}^2}=1\text{ and }\frac{\text{x}^2}{\text{A}^2}-\frac{\text{y}^2}{\text{B}^2}=1$
→
Find the intervals in which the following functions are increasing or decreasing.
$f(x) = (x - 1)(x - 2)^2$
→
Show that the curves $4x = y^2 $ and $4xy = k$ cut at right angles, if $k^2 = 512.$
→