JAMIA 2024 — Mathematics PYQ

Step 1: Simplify the Equation

Given:

Taking the inverse secant of both sides:

Since $a$ is a constant, ${\sec }^{-1}(a)$ is also a constant. Let's call this constant $k$:

Step 2: Express $y$ in terms of $x$

Rearrange the equation to isolate $y$:

Group the $x$ and $y$ terms:

Step 3: Differentiate with respect to $x$

Now, differentiate both sides with respect to $x$:

Since $\left(\frac{1-k}{1+k}\right)$ is a constant:

Substitute $k=\frac{x-y}{x+y}$ back into the expression:

Alternatively, from the relation $y=mx$ (where $m$ is a constant), we know that $\frac{y}{x}=m$. Therefore:

Final Answer

The derivative is: