1. What is the derivative of $\cos ^{2} 3 x$ with respect to $x$ ?

Step 1 ：We are given the function \(f = \cos^2(3x)\) and we are asked to find its derivative with respect to \(x\).

Step 2 ：We can use the chain rule to find the derivative of a composite 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.

Step 3 ：In this case, the outer function is \(\cos^2(u)\) and the inner function is \(3x\).

Step 4 ：The derivative of \(\cos^2(u)\) with respect to \(u\) is \(-2\cos(u)\sin(u)\) and the derivative of \(3x\) with respect to \(x\) is \(3\).

Step 5 ：By applying the chain rule, we get the derivative of \(f\) with respect to \(x\) as \(-6\sin(3x)\cos(3x)\).

Step 6 ：So, the derivative of \(\cos^2(3x)\) with respect to \(x\) is \(\boxed{-6\sin(3x)\cos(3x)}\).