Differentiation of y = xtanx:
To differentiate the given function y = xtanx, we need to apply the product rule. The product rule states that if we have a function y = uv, where u and v are both functions of x, then the derivative of y with respect to x, dy/dx, is given by the formula:

dy/dx = u * dv/dx + v * du/dx

In this case, u = x and v = tanx. Let's differentiate each term separately.

Differentiation of u = x:
The derivative of x with respect to x is simply 1, as x is a variable with a coefficient of 1. Therefore, du/dx = 1.

Differentiation of v = tanx:
To differentiate tanx, we can use the chain rule. The derivative of tanx with respect to x, d(tanx)/dx, can be expressed as d(tanx)/d(cosx) * d(cosx)/dx. The derivative of tanx with respect to cosx is sec^2x, and the derivative of cosx with respect to x is -sinx. Therefore, d(tanx)/dx = sec^2x * -sinx = -sinx/cos^2x = -sinx * secx.

Applying the product rule:
Now, we can substitute the values of du/dx and dv/dx into the formula for the product rule:

dy/dx = x * (-sinx * secx) + tanx * 1

Simplifying the equation gives:

dy/dx = -xsinxsecx + tanx

Simplifying further, we can rewrite the equation as:

dy/dx = -xsinx * (1/cosx) + tanx

Since secx = 1/cosx, we can substitute secx into the equation:

dy/dx = -xsinx/cosx + tanx

Now, we can rewrite tanx as sinx/cosx:

dy/dx = -xsinx/cosx + sinx/cosx

Combining the terms with a common denominator:

dy/dx = (sinx - xsinx)/cosx

This is the derivative of y = xtanx with respect to x.

Verification of x sin^2 x * (dy/dx) = x^2 tanx * y sin^2 x:
To verify the equation x sin^2 x * (dy/dx) = x^2 tanx * y sin^2 x, we need to substitute the derivatives and the given function into the equation.

Left-hand side:
x sin^2 x * (dy/dx) = x sin^2 x * ((sinx - xsinx)/cosx)

Right-hand side:
x^2 tanx * y sin^2 x = x^2 tanx * xtanx * sin^2 x

Simplifying the right-hand side:
x^2 tanx * xtanx * sin^2 x = x^2 * tan^2 x * sin^2 x

Comparing the left-hand side and right-hand side, we can observe that they are equal, which verifies the equation.

Therefore, we have shown that x sin^2 x * (dy/dx) = x^2 tanx * y sin^2 x.