Download App

Higher Order Derivatives — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSHigher Order Derivatives5 Marks
Question
If $\text{y}\log(1+\cos\text{x}),$ prove that $\frac{\text{d}^3\text{y}}{\text{dx}^3}+\frac{\text{d}^\text{y}}{\text{dx}^2}.\frac{\text{d}\text{y}}{\text{dx}}=0$
✓

Answer

$\text{y}\log(1+\cos\text{x}),$
Differentiating w.r.t.x, we get
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{1}{1+\cos\text{x}}\times-\sin\text{x}=\frac{-\sin\text{x}}{1+\cos\text{x}}$
Differentiating w.r.t.x, we get
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}=-\Big[\frac{(1+\cos\text{x})\cos\text{x}-\sin(-\sin\text{x})}{(1+\cos\text{x})^2}\Big]$
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}=-\Big[\frac{\cos\text{x}+\cos^2\text{x}+\sin^2\text{x}}{(1+\cos\text{x})^2}\Big]=-\Big[\frac{1+\cos\text{x}}{(1+\cos\text{x})^2}\Big]=\frac{-1}{1+\cos\text{x}}$
Differentiating w.r.t.x
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^3}=-\Big(\frac{+1}{(1+\cos\text{x})^2}\times+\sin\text{x}\Big)\\=-\Big(\frac{-\sin\text{x}}{1+\cos\text{x}}\Big)\times\Big(\frac{-1}{1+\cos\text{x}}\Big)=-\frac{\text{dy}}{\text{dx}}.\frac{\text{d}^2\text{y}}{\text{dx}^2}$
$\Rightarrow\frac{\text{d}^3\text{y}}{\text{dx}^3}+\frac{\text{d}^2\text{y}}{\text{dx}^2}.\frac{\text{dy}}{\text{dx}}=0$

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 Higher Order Derivatives→Higher Order Derivatives — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find the area of region bounded by the line $x = 2$ and the parabola $y^2 = 8x.$
→
Discuss the continuity of the following functions at the indicated point:
$\text{f}\text{(x)}=\begin{cases}\frac{2\text{x}+\text{x}^2}{\text{x}}, & \text{x} \neq0\\0,&\text{ x} = 0\end{cases}\text{at x}=0$
→
Find the mean and variance of the number of tails in three tosses of a coin.
→
A firm manufactures $3$ products $A, B$ and $C.$ The profits are $Rs. 3, Rs. 2$ and $Rs. 4$ respectively. The firm has $2$ machines and below is the required processing time in minutes for each machine on each product:
Machine
Products
$A$ $B$ $C$
$M_1$ $4$ $3$ $5$
$M_2$ $2$ $2$ $4$
Machines $M_1$ and $M_{2 }$ have $2000$ and $2500$ machine minutes respectively. The firm must manufacture $100 A\ 's, 200 B\ 's$ and $50 C\ 's$ but not more than $150 A\ 's.$ Set up a $\text{LPP}$ to maximize the profit.
→
A biased die is such that $\text{P}(4)=\frac{1}{10}$ and other scores being equally likely. The die is tossed twice. If X is the ‘number of fours seen’, find the variance of the random variable X.
→
Evaluate the following integrals:$\int(\log\text{x})^2\cdot\text{x dx}$
→
Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\cos^5\text{x dx}$
→
Give examples of two surjective functions $f_1$ and $f_2$ from $Z$ to $Z$ such that $f_1 + f_2$ is not surjective.
→
Evaluate: $\int\frac{\text{x}^{2}}{(\text{x}^{2} + 4)(\text{x}^{2} + 9 )}\text{dx}.$
→
A and B throw a pair of dice alternately. A wins the game if he gets a total of 6 and B wins if she gets a total of 7. It A starts the game, find the probability of winning the game by A in third throw of the pair of dice.
→