Download App

CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY4 Marks
Question
Differentiate the following functions with respect to x:
$\frac{3\text{x}^2\sin\text{x}}{\sqrt{7-\text{x}^2}}$
✓

Answer

Let $\text{y}=\frac{3\text{x}^2\sin\text{x}}{\sqrt{7-\text{x}^2}}$
Differentiate it with respect to x,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\bigg\{\frac{3\text{x}^2\sin\text{x}}{(\sqrt{7-\text{x}^2})^\frac{1}{2}}\bigg\}$
$=\frac{(7-\text{x}^2)^\frac{1}{2}\times\frac{\text{d}}{\text{dx}}(3\text{x}^3\sin\text{x})-(3\text{x}^2\sin\text{x})\frac{\text{d}}{\text{dx}}(7-\text{x}^2)^\frac{1}{2}}{\Big[(7-\text{x}^2)^\frac{1}{2}\Big]^2}$
[Using quotient rule, chain rule and product rule]
$\Bigg[\frac{(7-\text{x}^2)^\frac{1}{2}\times3\Big[\text{x}^2\frac{\text{d}}{\text{dx}}(\sin\text{x})+\sin\text{x}\frac{\text{d}}{\text{dx}}(\text{x}^2)\Big]-3\text{x}^2\sin\text{x}\times\frac{1}{2}(7-\text{x}^2)\times\frac{\text{d}}{\text{dx}}(7-\text{x}^2)}{(7-\text{x}^2)}\Bigg]$
$\Bigg[\frac{(7-\text{x}^2)^\frac{1}{2}3(\text{x}^2\cos\text{x}+2\text{x}\sin\text{x})-3\text{x}^2\sin\text{x}\times\frac{1}{2}(7-\text{x}^2)^\frac{-1}{2}(-2\text{x})}{(7-\text{x}^2)}\Bigg]$
$=\Bigg[\frac{(7-\text{x}^2)^\frac{1}{2}\times3(\text{x}^2\cos+2\text{x}\sin\text{x})}{(7-\text{x}^2)}+\frac{3\text{x}^2\sin\text{x}(7-\text{x}^2)^\frac{-1}{2}}{(7-\text{x}^2)}\Bigg]$
$\bigg[\frac{6\text{x}\sin\text{x}+3\text{x}^2\cos\text{x}}{\sqrt{(7-\text{x}^2)}}+\frac{3\text{x}^3\sin\text{x}}{(7-\text{x}^2)^\frac{3}{2}}\bigg]$
So,
$\frac{\text{d}}{\text{dx}}\Big(\frac{3\text{x}^2\sin\text{x}}{\sqrt{7-\text{x}^2}}\Big)\bigg[\frac{6\text{x}\sin\text{x}+3\text{x}^2\cos\text{x}}{\sqrt{(7-\text{x}^2)}}+\frac{3\text{x}^3\sin\text{x}}{(7-\text{x}^2)^\frac{3}{2}}\bigg]$

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

Find the points of discontinuity, if any of the following function:
$\text{f(x)}=\begin{cases}\frac{\text{x}^4-16}{\text{x}-2},&\text{if }\text{ x}\neq2\\16,&\text{if }\text{ x}=2\end{cases}$
→
Determine graphically the minimum value of the objective function Z = -50x + 20y subject to the constraints:
2x - y $\geq$ - 5
3x + y $\geq$ 3
2x - 3y $\leq$ 12
x $\geq$ 0, y $\geq$ 0
→
Determine the equation of the line passing through the points (1, 2, -4) and perpendicular to the lines $\frac{\text{x}-8}{8}=\frac{\text{y}+9}{-16}=\frac{\text{z}-10}{7}$ and $\frac{\text{x}-15}{3}=\frac{\text{y}-29}{8}=\frac{\text{z}-5}{-5}.$
→
Three urns A, B and C contain 6 red and 4 white; 2 red and 6 white; and 1 red and 5 white balls respectively. An urn is chosen at random and a ball is drawn. If the ball drawn is found to be red, find the probability that the ball was drawn from urn A.
→
Solve the following differential equation:
$(1+\text{x}^2)\frac{\text{dy}}{\text{dx}}-2\text{xy}=(\text{x}^2+2)(\text{x}^2+1)$
→
Solve the following initial value problems:
$\text{x}(\text{x}^2+3\text{y}^2)\text{dx}+\text{y}(\text{y}^2+3\text{x}^2)\text{dy}=0,\text{y}(1)=1$
→
Find the shortest distance between the lines:
$\vec{\text{r}}=(\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}})+\lambda(\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}})\ \text{and}\ $
$\vec{\text{r}}=2\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}+\mu(2\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}})$
→
Discuss the continuity of the following functions:
  1. $\text{f(x)}=\sin\text{x}+\cos\text{x}$
  2. $\text{f(x)}=\sin\text{x}-\cos\text{x}$
  3. $\text{f(x)}=\sin\text{x}\cos\text{x}$
→
Solve the following differential equation:
$\text{y e}^{\frac{\text{x}}{\text{y}}}\text{dx}=\big(\text{xe}^{\frac{\text{x}}{\text{y}}}+\text{y}\big)\text{dy}$
→
Compute the area bounded by the lines x + 2y = 2, y - x = 1 and 2x + y = 7.
→