𝜕𝜕𝑥 (2 tan−1 (𝑦𝑥))

To find the derivative of the function with respect to x, we will use the chain rule and the derivative of the inverse tangent function.

The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

The derivative of the inverse tangent function is 1/(1+x^2).

So, let's start:

1. Differentiate the outer function: The derivative of 2 tan^(-1)(y/x) with respect to (y/x) is 2/(1+(y/x)^2).

2. Differentiate the inner function: The derivative of (y/x) with respect to x is -y/x^2.

3. Apply the chain rule: Multiply the results from steps 1 and 2 to get the derivative of the original function with respect to x.

So, the derivative of 2 tan^(-1)(y/x) with respect to x is -2y/(x^2(1+(y/x)^2)).