**Differentiation of y = tan(x y)**

To find the derivative of the function y = tan(xy), we will use the chain rule. The chain rule is a method for finding the derivative of composite functions, which are functions that can be expressed as the composition of two or more functions.

**Using the Chain Rule:**

The chain rule states that if we have a composite function f(g(x)), then the derivative of f(g(x)) with respect to x is given by:

(dy/dx) = (dy/du) * (du/dx)

Where u = g(x) and y = f(u).

**Applying the Chain Rule:**

In our case, we have the composite function y = tan(xy), where u = xy and y = tan(u). Let's differentiate each part separately:

1. Differentiating y = tan(u) with respect to u:
- The derivative of tan(u) with respect to u is sec^2(u). This can be derived from the basic trigonometric identity: d/dx(tan(x)) = sec^2(x).

2. Differentiating u = xy with respect to x:
- To differentiate u = xy, we treat y as a constant because we are differentiating with respect to x. So, the derivative of u with respect to x is simply y.

Now, we can apply the chain rule:

(dy/dx) = (dy/du) * (du/dx)
= sec^2(u) * y

Substituting u = xy, we get:

(dy/dx) = sec^2(xy) * y

Therefore, the derivative of y = tan(xy) with respect to x is sec^2(xy) * y.

**Summary:**

The derivative of y = tan(xy) with respect to x is given by dy/dx = sec^2(xy) * y. We used the chain rule to differentiate the composite function y = tan(xy) by differentiating the inner function and the outer function separately. Finally, we substituted the value of the inner function (u = xy) into the derivative expression.