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

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 "4 Marks" 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

Solve the system of linear equation, using matrix method 2x + y + z = 1; $x - 2y - z = \frac{3}{2};\,\,3y - 5z = 9$

→

Evaluate the following intregals:
$\int\frac{\text{x}+1}{\sqrt{\text{x}^2+1}}\text{dx}$

→

Evaluate the following integrals:
$\int^\limits{\frac{\pi}{2}}_{0}\sqrt{\cos\text{x}-\cos^3\text{x}}(\sec^3\text{x}-1)\cos^2\text{x dx}$

→

Find the shortest distance between the following pairs of parallel lines whose equations are:

$\vec{\text{r}}=\big(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}\big)+\lambda\big(\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}\big)$ and $\vec{\text{r}}=\big(2\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}\big)+\mu\big(-\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}\big)$

→

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.
f(x) = x(x - 1) on [1, 2]

→

If the area bounded by the parabola $\text{y}^{2} = 16\text{ax}$ and the line $\text{y = 4 mx}$ is $\frac{\text{a}^{2}}{12}$ sq. units, then using integration, find the value of m.

→

Find the equations of the two lines through the origin which intersect the line $\frac{\text{x}-3}{2}-\frac{\text{y}-3}{1}=\frac{\text{z}}{1}$ at angles of $\frac{\pi}{3}$ each.

→

Find the approximate change in the surface area of a cube of side x metres caused by decreasing the side by 1%.

→

Find the equation of a plane passing through the points (0, 0, 0) and (3, -1,2) and parallel to the line $\frac{\text{x}-4}{1}=\frac{\text{y}+3}{-4}=\frac{\text{z}+1}{7}$

→

Evaluate the following integrals:
$\int\limits^{\text{a}}_{-\text{a}}\log\Big(\frac{\text{a}-\sin\theta}{\text{a}+\sin\theta}\Big)\text{d}\theta$