To find the derivative of \(y = \sin^{2} (5x)\) with respect to x, we can use the chain rule. The chain rule states that if we have a function \(f(g(x))\), then its derivative is \(f'(g(x)) \cdot g'(x)\). In this case, our outer function is \(f(u) = u^{2}\) and our inner function is \(g(x) = \sin(5x)\). The derivative of the outer function is \(f'(u) = 2u\) and the derivative of the inner function is \(g'(x) = 5\cos(5x)\). Applying the chain rule, we get:

\(\frac{dy}{dx} = f'(g(x)) \cdot g'(x) = 2\sin(5x) \cdot 5\cos(5x) = 10\sin(5x)\cos(5x)\)

So the derivative of \(y = \sin^{2} (5x)\) with respect to x is 10 sin 5x cos 5x