Download App

Higher Order Derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsHigher Order Derivatives5 Marks
Question
If $\text{x}=\text{a}(\cos\text{t}+\log\tan\frac{\text{t}}{2})\ \text{and}\ \text{y}=\text{a}(\sin\text{t}),$evaluate $\frac{\text{d}^2\text{y}}{\text{dx}^2}\ \text{at}\ \text{t}=\frac{\pi}{3}.$
✓

Answer

We have,
$\text{x}=\text{a}(\cos\text{t}+\log\tan\frac{\text{t}}{2})\ \text{and}\ \text{y}=\text{a}(\sin\text{t}),$
On differentiating with respect to t, we get
$\frac{\text{dx}}{\text{dt}}=\frac{\text{d}}{\text{dt}}\Big[\text{a}\Big(\cos\text{t}+\log\tan\frac{\text{t}}{2}\Big)\Big]=\text{a}\Bigg(-\sin\text{t}+\frac{1}{\tan\frac{\text{t}}{2}}\times\sec^2\frac{\text{t}}{2}\times\frac{1}{2}\Bigg)$
$=\text{a}\Bigg(-\sin\text{t}+\frac{1}{2\sin\frac{\text{t}}{2}\cos\frac{\text{t}}{2}}\Bigg)=\text{a}\Big(-\sin\text{t}+\frac{1}{\sin\text{t}}\Big)$
$=\text{a}\Big(\frac{-\sin^2\text{t}+1}{\sin\text{t}}\Big)=\text{a}\Big(\frac{\cos^2\text{t}}{\sin\text{t}}\Big)$
And
$\frac{\text{dy}}{\text{dt}}=\frac{\text{d}}{\text{dt}}(\text{a}\sin\text{t})=\text{a}\cos\text{t}$
Now, $\Big(\frac{\text{dy}}{\text{dx}}\Big)=\frac{\frac{\text{dy}}{\text{dt}}}{\frac{\text{dX}}{\text{dt}}}=\frac{\text{a}\cos\text{t}}{\text{a}\frac{\cos^2\text{t}}{\sin\text{t}}}=\tan\text{t}$
Therefore
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{\text{d}}{\text{dx}}\Big(\frac{\text{dy}}{\text{dx}}\Big)=\frac{\text{d}}{\text{dx}}(\tan(\text{t}))$
$=\frac{\text{d}}{\text{dt}}(\tan(\text{t}))\times\frac{\text{dt}}{\text{dx}}=\sec^2\text{t}\times\frac{\sin\text{t}}{\text{a}\cos^2\text{t}}$
$=\Big(\frac{\sin\text{t}}{\text{a}\cos^4\text{t}}\Big)$
$\Big(\frac{\text{d}^2\text{y}}{\text{dx}^2}\Big)_{\text{t}=\frac{\pi}{3}}=\Bigg(\frac{\sin\Big(\frac{\pi}{3}\Big)}{\text{a}\cos^4\Big(\frac{\pi}{3}\Big)}\Bigg)=\frac{\frac{\sqrt{3}}{2}}{\text{a}\Big(\frac{1}{16}\Big)}=\frac{8\sqrt{3}}{\text{a}}$
Hence, $\text{at}\ \text{t}=\frac{\pi}{3},\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{8\sqrt{3}}{\text{a}}$

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

Differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}-3\frac{\text{dy}}{\text{dx}}+2\text{y}=0,\text{y}(0)=1,\text{y}(0)=3$
Function $\text{y}=\text{e}^\text{x}+\text{e}^{2\text{x}}$
→
Find the distance of the point (1, -2, 4) from plane passing throuhg the point (1, 2, 2) and perpendicular of the planes x - y + 2z = 3 and 2x - 2y + z + 12 = 0
→
If $\text{x}=\cos\text{t}+\log\tan\Big(\frac{\text{t}}{2}\Big),\ \text{y}=\sin\text{t},$ then find the values of $\frac{\text{d}^2\text{y}}{\text{dt}^2}$ and $\frac{\text{d}^2\text{y}}{\text{dx}^2}$ at $\text{t}=\frac{\pi}{4}.$
→
Differentiate $\sin^{-1}\Big(4\text{x}\sqrt{1-4\text{x}^2}\Big)$ with respect to $\sqrt{1-4\text{x}^2},$ if:
$\text{x}\in\Big(-\frac{1}{2\sqrt{2}},\frac{1}{2}\Big)$
→
Prove that the line through A(0, –1, –1) and B(4, 5, 1) intersects the line through C(3, 9, 4) and D(–4, 4, 4).
→
Solve the following differential equations:
$\frac{\text{dy}}{\text{dx}}=\frac{\text{y}^2-\text{x}^2}{2\text{xy}}$
→
Two cards are drawn simultaneosly from a well shuffled deck of 52 cards. Find the probability distribution of the number of the successes, when getting a spade is considered a success.
→
Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{\cos\text{x}+\sin\text{x}}{\cos\text{x}-\sin\text{x}}\Big), \frac{\pi}{4}<\text{x}<\frac{\pi}{4}$
→
If $\log\sqrt{\text{x}^2+\text{y}^2}=\tan^{-1}\Big(\frac{\text{y}}{\text{x}}\Big),$ prove that $\frac{\text{dx}}{\text{dx}}=\frac{\text{x}+\text{y}}{\text{x}-\text{y}}$
→
Solve the following differential equation:
$\big(\cot^{-1}\text{y} + \text{x}\big)\text{dy}= \big(1 + \text{y}^2\big) \text{dx}$
→