Download App

Higher Order Derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsHigher Order Derivatives5 Marks
Question
If $\text{y}=\text{cosec}^{-1}\text{x},\text{x}>1$ prove that $\text{x}(\text{x}^2-1)\frac{\text{d}^2\text{y}}{\text{dx}^2}+(2\text{x}^2-1)\frac{\text{dy}}{\text{dx}}=0.$
✓

Answer

Here,
$\text{y}=\text{cosec}^{-1}\text{x}$
Differentiating w.r.t.x, we get
$\frac{\text{dy}}{\text{dx}}=\frac{-1}{\text{x}\sqrt{{}\text{x}^2-1}}$
Differentiating w.r.t.x, we get
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{\sqrt{\text{x}^2-1}+\frac{\text{x}^2}{\sqrt{\text{x}^2-1}}}{\text{x}^2(\text{x}^2-1)}$
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{\text{x}^2-1+\text{x}^2}{\text{x}^2(\text{x}^2-1)\sqrt{\text{x}^2-1}}$
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{2\text{x}^2-1}{\text{x}^2(\text{x}^2-1)\sqrt{\text{x}^2-1}}$
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{2\text{x}}{(\text{x}^2-1)\sqrt{\text{x}^2-1}}-\frac{1}{\text{x}^2(\text{x}^2-1)\sqrt{\text{x}^2-1}}$
$\Rightarrow(\text{x}^2-1)\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{2\text{x}}{\sqrt{\text{x}^2-1}}-\frac{1}{\sqrt{\text{x}^2-1}}$
$\Rightarrow(\text{x}^2-1)\frac{\text{d}^2\text{y}}{\text{dx}^2}=-2\text{x}\frac{\text{dy}}{\text{dx}}+\frac{1}{\text{x}}\frac{\text{dy}}{\text{dx}}$
$\Rightarrow(\text{x}^2-1)\frac{\text{d}^2\text{y}}{\text{dx}^2}=-2\text{x}\frac{\text{dy}}{\text{dx}}+\frac{1}{\text{x}}\frac{\text{dy}}{\text{dx}}$
$\Rightarrow(\text{x}^2-1)\frac{\text{d}^2\text{y}}{\text{dx}^2}=-(2\text{x}^2-1)\frac{\text{dy}}{\text{dx}}$
$\Rightarrow(\text{x}^2-1)\frac{\text{d}^2\text{y}}{\text{dx}^2}=+(2\text{x}^2-1)\frac{\text{dy}}{\text{dx}}=0$
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 Higher Order Derivatives→Higher Order Derivatives — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluate the following intregals:
$\int\frac{\text{x}}{(\text{x}+1)(\text{x}^2+1)}\ \text{dx}$
→
A and B take turns in throwing two dice, the first to throw 9 being awarded the prize. Show that their chance of winning are in the ratio 9 : 8.
→
Let $\text{A}=\begin{bmatrix}3 & 2 \\7 & 5 \end{bmatrix}\text{and B}=\begin{bmatrix}6 & 7 \\8 & 9 \end{bmatrix}$. Find $(AB)^{-1}$.
→
A bag contains 25 tickets, numbered from 1 to 25. A ticket is drawn and then another ticket is drawn without replacement. Find the probability that both tickets will show even numbers.
→
Find the vector equations of the following planes in scalar product form $(\vec{\text{r}}\cdot\vec{\text{n}}=\text{d}):$
$\vec{\text{r}}=(1+\text{s}-\text{t})\hat{\text{t}}+(2-\text{s})\hat{\text{j}}+(3-2\text{s}+2\text{t})\hat{\text{k}}$
→
Find which of the function:
$\text{f(x)}=\begin{cases}|\text{x}|\cos\frac{1}{\text{x}},&\text{if x}\neq0\\0,&\text{if x}=0\end{cases}$
at x = 0
→
If R is a relation on the set A = {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3)}, then R is:
  1. Reflexive.
  2. Symmetric.
  3. Transitive.
  4. All the three options.
→
If $\text{f}\text{(x)}=\begin{cases}\frac{1-\cos\text{x}}{\text {x}^2}, & \text{when} \text{ x}\neq 0\\1, & \text{when}\text{ x} = 0\end{cases}$ Show that f(x) is discontinuous at x = 0.
→
Prove that the diagonals of a rhombus are perpendicular bisectors of each other.
→
There are three coins. One is a two-headed coin (having head on both faces), another is a biased coin that comes up heads $75\%$ of the times and third is also a biased coin that comes up tails $40\%$ of the times. One of the three coins is chosen at random and tossed, and it shows heads. What is the probability that it was the two-headed coin?
→