If ( )^y=( )^x, find dy dx - Vidyadip

Question

If $(\cos\text{x})^\text{y}=(\cos\text{y})^\text{x},$ find $\frac{\text{dy}}{\text{dx}}$

✓

Answer

$(\cos\text{x})^\text{y}=(\cos\text{y})^\text{x}$
Taking log on both sides we get
$\text{y}\log\cos\text{x}=\text{x}\log\cos\text{y}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}\log\cos\text{x}-\text{y}\tan\text{x}=\log\cos\text{y}-\text{x}\tan\text{y}\frac{\text{dy}}{\text{dx}}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}\log\cos\text{x}+\text{x}\tan\text{y}\frac{\text{dy}}{\text{dx}}=\log\cos\text{y}+\text{y}\tan\text{x}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}(\log\cos\text{x}+\text{x}\tan\text{y})=\log\cos\text{y}+\text{y}\tan\text{x}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{\log\cos\text{y}+\text{y}\tan\text{x}}{\log\cos\text{x}+\text{x}\tan\text{y}}$

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

Similar questions

Find the angle between the follwing pairs of lines:$\vec{\text{r}}=\lambda\big(\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}}\big)$ and $\vec{\text{r}}=2\hat{\text{j}}+\mu\big\{\big(\sqrt{3}-1\big)\hat{\text{i}}-\big(\sqrt{3}+1\big)\hat{\text{j}}+4\hat{\text{k}}\big\}$

Find the shortest distance between the following pairs of lines whose vector equation are:
$\vec{\text{r}}=\big(\hat{\text{i}}+\hat{\text{j}}\big)+\lambda\big(2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}\big)$ and $\vec{\text{r}}=2\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}+\mu\big(3\hat{\text{i}}-5\hat{\text{j}}+2\hat{\text{k}}\big)$

The function $\text{f(x)}=\begin{cases}\frac{\text{x}^2}{\text{a}},&\text{if }0\leq\text{ x}<1\\\text{a},&\text{if }1\leq\text{x}<\sqrt{2}\\\frac{2\text{b}^2-4\text{b}}{\text{x}^2},&\text{if }\sqrt{2}\leq\text{x}<\infty\end{cases}$ is continuous on $(0,\infty),$ then find the most suitable value of a and b.

Differentiate the following functions with respect to x:
$\log\Big(\frac{\text{x}^2+\text{x}+1}{\text{x}^3-\text{x}+1}\Big)$

If for function $\phi(\text{x})=\lambda\text{x}^2+7\text{x}-4, \phi(5)=97,$ find $\lambda.$

Find the area of the region bounded by the curve $\text{y}=\sqrt{1-\text{x}^2},$ line y = x and the positive x-axis.

Find the mean variance and standard deviation of the following probability distribution

$x_i$	$a$	$b$
$p_i$	$p$	$q$

Where $p + q = 1$

A producer has 30 and 17 units of labour and capital respectively which he can use to produce two type of goods x and y. To produce one unit of x, 2 units of labour and 3 units of capital are required. Similarly, 3 units of labour and 1 unit of capital is required to produce one unit of y. If x and y are priced at Rs. 100 and Rs. 120 per unit respectively, how should be producer use his resources to maximize the total revenue? Solve the problem graphically.

Find a vector of magnitude 26 units normal to the plane 12x - 3y + 4z = 1

$\overrightarrow{\text{AB}}=3\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}$ and $\overrightarrow{\text{CD}}=-3\hat{\text{i}}+2\hat{\text{j}}+4\hat{\text{k}}$ are two vectors. The position vectors of the points A and C are $6\hat{\text{i}}+7\hat{\text{j}}+4\hat{\text{k}}$ and $-9\hat{\text{j}}+2\hat{\text{k}},$ respectively. Find the position vector of a point P on the line AB and a point Q on the line CD such that $\overrightarrow{\text{PQ}}$ is perpendicular to $\overrightarrow{\text{AB}}$ and $\overrightarrow{\text{CD}}$ both.