Download App

CONTINUITY AND DIFFERENTIABILITY — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=(\tan\text{x})^{\cot\text{x}}+(\cot\text{x})^{\tan\text{x}}$
✓

Answer

Here,
$\text{y}=(\tan\text{x})^{\cot\text{x}}+(\cot\text{x})^{\tan\text{x}}$
$\text{y}=\text{e}^{\log(\tan\text{x})^{\cot\text{x}}}+\text{e}^{\log(\cot\text{x})^{\tan\text{x}}}$
$\big[\text{Since},\log_\text{e}\text{e}=1,\log\text{a}^\text{b}=\text{b}\log\text{a}\big]$
$\text{y}=\text{e}^{\cot\text{x}\log\tan\text{x}}+\text{e}^{\tan\text{x}\log(\cot\text{x})}$
Differentiating it with respect to x using rule and product rule,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\big(\text{e}^{\cot\text{x}\log\tan\text{x}}\big)+\frac{\text{d}}{\text{dx}}\big(\text{e}^{\tan\text{x}\log\cot\text{x}}\big)$
$=\text{e}^{\cot\text{x}\log\tan\text{x}}\frac{\text{d}}{\text{dx}}(\cot\text{x}\log\tan\text{x})+\text{e}^{\tan\text{x}\log\cot\text{x}}\frac{\text{d}}{\text{dx}}(\tan\text{x}\log\cot\text{x})$
$=\text{e}^{\log(\tan\text{x})^{\cot\text{x}}}\Big[\cot\text{x}\frac{\text{d}}{\text{dx}}\log\tan\text{x}+\log\tan\text{x}\frac{\text{d}}{\text{dx}}\cot\text{x}\Big] \\ +\text{e}^{\log(\cot\text{x})^{\tan\text{x}}}\Big[ \tan\text{x}\frac{\text{d}}{\text{dx}} \log\cot\text{x}+\log\cot\text{x}\frac{\text{d}}{\text{dx}}(\tan\text{x})\Big]$
$=(\tan\text{x})^{\cot\text{x}}\Big[\cot\text{x}\times\Big(\frac{1}{\tan\text{x}}\Big)\frac{\text{d}}{\text{dx}}(\tan\text{x})+\log\tan\text{x}(-\text{cosec}^2\text{x})\Big] \\ +(\cot\text{x})^{\tan\text{x}}\Big[\tan\text{x}\big(\frac{1}{\cot\text{x}}\big)\frac{\text{d}}{\text{dx}}(\cot\text{x})+\log\cot\text{x}\big(\sec^2\text{x}\big)\Big]$
$=(\tan\text{x})^{\cot\text{x}}\Big[(1)\big(\sec^2\text{x}\big)-\text{cosec}^2\text{x}\log\tan\text{x}\Big] \\ +(\cot)^{\tan\text{x}}\Big[(1)\big(-\text{cosec}^2\text{x}\big)+\sec^2\text{x}\log\cot\text{x}\Big]$
$\frac{\text{dy}}{\text{dx}}=(\tan\text{x})^{\cot\text{x}}\Big[\sec^{2\text{x}}-\text{cosec}^2\text{x}\log\tan\text{x}\Big] \\ +(\cot)^{\tan\text{x}}\Big[\sec^2\text{x}\log\cot\text{x}-\text{cosec}^2\text{x}\Big]$

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

Evaluate the following integrals:
$\int\limits^{5}_0\frac{\sqrt[4]{\text{x}+4}}{\sqrt[4]{\text{x}+4}+\sqrt[9]{9-\text{x}}}\text{ dx}$
→
If $\text{A}=\begin{bmatrix}2&-2\\4&2\\-5&1\end{bmatrix},\text{ B}=\begin{bmatrix}8&0\\4&-2\\3&6\end{bmatrix},$ find matrix X such that 2A + 3X = 5B.
→
Show that the following systems of linear equations is inconsistent:
3x - y + 2z = 6,
2x - y + z = 2,
3x + 6y + 5z = 20
→
Determine the values of $x$ for which the function $f(x) = x^2 - 6x + 9$ is increasing or decreasing. Also, find the coordinates of the point on the curve $y = x^2 - 6x + 9$ where the normal is parallel to the line $y = x + 5.$
→
Find the distance of the point (3, 3, 3) from the plane $\vec{\text{r}}\cdot(5\hat{\text{i}}+2\hat{\text{j}}-7\hat{\text{k}})+9=0$
→
Solve the following differential equations:$(\text{xy}^2+2\text{x})\text{dx}+(\text{x}^2\text{y+2y})\text{dy}=0$
→
A bag contains $4$ white and $5$ black balls and another bag contains $3$ white and $4$ black balls. A ball is taken out from the first bag and without seeing its colour is put in the second bag. A ball is taken out from the latter. Find the probability that the ball drawn is white.
→
Find the angle between the follwing pairs of lines : $\vec{\text{r}}=\big(3\hat{\text{i}}+2\hat{\text{j}}-4\hat{\text{k}}\big)+\lambda\big(\hat{\text{i}}+2\hat{\text{j}}+2\hat{\text{k}}\big)$ and $\vec{\text{r}}=\big(5\hat{\text{j}}-2\hat{\text{k}}\big)+\mu\big(3\hat{\text{i}}+2\hat{\text{j}}+6\hat{\text{k}}\big)$
→
Evaluate the following integrals as limit of sum:
$\int\limits^2_{1}\big(\text{x}^2-1\big)\text{dx}$
→
There are three coins. One is two-headed coin (having head on both faces), another is biased coin that comes up heads 75% of the times and third is also a biased coin that comes up tail 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?
→