If (x+y)+ (x+y)=1, find dy dx - Vidyadip

Question

If $\tan(\text{x}+\text{y})+\tan(\text{x}+\text{y})=1,$ find $\frac{\text{dy}}{\text{dx}}$

✓

Answer

We have, $\tan(\text{x}+\text{y})+\tan(\text{x}-\text{y})=1$
Differentiating with respect to x, we get,
$\Rightarrow\ \frac{\text{d}}{\text{dx}}\tan(\text{x}+\text{y})+\frac{\text{d}}{\text{dx}}\tan(\text{x}+\text{y})=\frac{\text{d}}{\text{dx}}(1)$
$\Rightarrow\ \sec^2(\text{x}+\text{y})\frac{\text{d}}{\text{dx}}(\text{x}+\text{y})+\sec^2(\text{x}-\text{y})\frac{\text{d}}{\text{dx}}(\text{x}-\text{y})=0$
$\Rightarrow\ \sec^2(\text{x}+\text{y})\Big[1+\frac{\text{dy}}{\text{dx}}\Big]+\sec^2(\text{x}-\text{y})\Big[1-\frac{\text{dy}}{\text{dx}}\Big]=0$
$\Rightarrow\ \sec^2(\text{x}+\text{y})\frac{\text{dy}}{\text{dx}}-\sec^2(\text{x}-\text{y})\frac{\text{dy}}{\text{dx}} \\ =-\big[\sec^2(\text{x}-\text{y})+\sec^2(\text{x}-\text{y})\big]$
$\Rightarrow\frac{\text{dy}}{\text{dx}}\big[\sec^2(\text{x}+\text{y})-\sec^2(\text{x}-\text{y})\big] \\ =-\big[\sec^2(\text{x}+\text{y})+\sec^2(\text{x}-\text{y})\big]$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{\sec^2(\text{x}+\text{y})+\sec^2(\text{x}-\text{y})}{\sec^2(\text{x}-\text{y})-\sec^2(\text{x}+\text{y})}$

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

Similar questions

Solve the following differential equation
$(\text{x}^2+1)\frac{\text{dy}}{\text{dx}}=1$

If $a_1, a_2, a_3, ..., a_n$ is an arithmetic progression with common difference $d,$ then evaluate the following expression.
$\tan\bigg[\tan^{-1}\Big(\frac{\text{d}}{1+\text{a}_1\text{a}_2}\Big)+\tan^{-1}\Big(\frac{\text{d}}{1+\text{a}_2\text{a}_3}\Big)+\tan^{-1}\Big(\frac{\text{d}}{1+\text{a}_3\text{a}_4}\Big)+\ .....\ +\tan^{-1}\Big(\frac{\text{d}}{1+\text{a}_{\text{n}-1}\text{a}_\text{n}}\Big)\bigg]$

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_{0}\frac{\tan\text{x}}{1+\text{m}^2\tan^2\text{x}}\text{ dx}$

Find the shortest distance between the following pairs of lines whose vector equation are:
$\vec{\text{r}}=(\lambda-1)\hat{\text{i}}+(\lambda+1)\hat{\text{j}}-(1+\lambda)\hat{\text{k}}$ and $\vec{\text{r}}=(1-\mu)\hat{\text{i}}+(2\mu-1)\hat{\text{j}}+(\mu+2)\hat{\text{k}}$

Divide $15$ into two parts such that the square of one multiplied with the cube of the other is minimum.

If $\text{f(x)}=\sqrt{1-\text{x}}$ and $\text{g(x)}=\log_\text{e}\text{x}$ are two real functions, then describe, functions fog and gof.

Find the shortest distance between the following pairs of lines whose vector equation are:
$\vec{\text{r}}=\big(3\hat{\text{i}}+5\hat{\text{j}}+7\hat{\text{k}}\big)+\lambda\big(\hat{\text{i}}-2\hat{\text{j}}+7\hat{\text{k}}\big)$ and $\vec{\text{r}}=-\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}+\mu\big(7\hat{\text{i}}-6\hat{\text{j}}+\hat{\text{k}}\big)$

Evaluate the following integrals:
$\int(4\text{x}+1)\sqrt{\text{x}^2-\text{x}-2}\text{dx}$

Find the equation of the plane passing through the point (2, 1, -1) and (-1, 3, 4) and perpendicular to the plane x - 2y + 4z = 10.

Evaluate the following integrals as limit of sum:
$\int\limits^{5}_{0}(\text{x}+1)\text{dx}$