Download App

CONTINUITY AND DIFFERENTIABILITY — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
If $\text{y}=\text{x}\sin(\text{a}+\text{y}),$ prove that $\frac{\text{dx}}{\text{dx}}=\frac{\sin^2(\text{a}+\text{y})}{\sin(\text{a}+\text{y})-\text{y}\cos(\text{a}+\text{y})}$
✓

Answer

We have, $\text{y}=\text{x}\sin(\text{a}+\text{y})$
Differentiate with respect to y,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\big[\text{x}\sin(\text{a}+\text{y})\big]$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{x}\frac{\text{d}}{\text{dx}}\{\sin(\text{a}+\text{y})\}+\sin(\text{a}+\text{y})\frac{\text{d}}{\text{dx}}(\text{x})$
[Using product rule and chain rule]
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{x}\cos(\text{a}+\text{y})\frac{\text{d}}{\text{dx}}+\sin(\text{a}+\text{y})(1)$
$\Rightarrow\frac{\text{dy}}{\text{dx}}\{1-\text{x}\cos(\text{a}+\text{y})\}=\sin(\text{a}+\text{y})$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{\sin(\text{a}+\text{y})}{1-\text{x}\cos(\text{a}+\text{y})}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{\sin(\text{a}+\text{y})}{1-\frac{\text{y}}{\sin(\text{a}+\text{y})}\cos(\text{a}+\text{y})}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{\sin^2(\text{a}+\text{y})}{\sin(\text{a}+\text{y}){-\text{y}\cos(\text{a}+\text{y})}}$
Hence, proved.

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→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Using integration, find the area of the region in the first quadrant enclosed by the $x-$axis, the line $y = x$ and the circle $x^2 + y^2 = 32.$
→
Differentiate the following functions from first principles:
$\text{e}^{\cos\text{x}}$
→
The probabilities of two students $A$ and $B$ coming to the school in time are $\frac{3}{7}$ and $ \frac{5}{7}$ respectively. Assuming that the events, ‘$A$ coming in time’ and ‘$B$ coming in time’ are independent, find the probability of only one of them coming to the school in time. Write at least one advantage of coming to school in time.
→
Prove the following identities:
$\begin{vmatrix}\text{a}^3&2&\text{a}\\\text{b}^3&2&\text{b}\\\text{c}^3&2&\text{c}\end{vmatrix}$
$=2(\text{a}-\text{b})(\text{b}-\text{c})(\text{c}-\text{a})(\text{a}+\text{b}+\text{c})$
→
 Prove that the area common to tha two parabola $y = 2x^2$ and $y = x^2 + 4$ is $323$ sq. units.
→
Two cards are drawn successively with replacement from well shuffled pack of 52 cards. Find the probability distribution of the number of kings.
→
A pair of dice is thrown. Let E be the event that the sum is greater than or equal to 10 and F be the event "5 appears on the first-die". Find $\text{P}\Big(\frac{\text{E}}{\text{F}}\Big)$. If F is the event "5 appears on at least one die", find $\text{P}\Big(\frac{\text{E}}{\text{F}}\Big)$.
→
If $\text{A}=\begin{bmatrix}3&1\\-1&2\end{bmatrix},$ show that $A^2 - 5A + 7I = 0$ use this to find $A^4.$
→
If y(x) is a solution of $\Big(\frac{2+\sin\text{x}}{1+\text{y}}\Big)\frac{\text{dy}}{\text{dx}}=-\cos\text{x}$ and y(0) = 1, then find the value of $\text{y}\big(\frac{\pi}{2}\big).$
→
Solve the following determinant equations:
$\begin{vmatrix}1&\text{x}&\text{x}^2\\1&\text{a}&\text{a}^2\\1&\text{b}&\text{b}^2\end{vmatrix}=0,\text{a}\neq\text{b}$
→