Problem:
Find dy/dx if y = 2 sin(3 cos 4x).

Solution:
To find dy/dx, we need to differentiate the given function y = 2 sin(3 cos 4x) with respect to x. Let's break down the process step by step.

Step 1: Use the Chain Rule
The given function y = 2 sin(3 cos 4x) involves composite functions. To differentiate it, we need to apply 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 f'(g(x)) multiplied by g'(x). In our case, f(x) = 2 sin(x) and g(x) = 3 cos(4x).

Step 2: Differentiate the Outer Function
First, let's find the derivative of the outer function f(x) = 2 sin(x). The derivative of sin(x) is cos(x), and since we have a constant multiple of 2, the derivative of 2 sin(x) is 2 cos(x).

Step 3: Differentiate the Inner Function
Now, let's find the derivative of the inner function g(x) = 3 cos(4x). The derivative of cos(x) is -sin(x), and again, we have a constant multiple of 3, so the derivative of 3 cos(4x) is -3 sin(4x).

Step 4: Apply the Chain Rule
Using the chain rule, we multiply the derivative of the outer function (2 cos(x)) by the derivative of the inner function (-3 sin(4x)). This gives us dy/dx = 2 cos(x) * (-3 sin(4x)).

Step 5: Simplify the Expression
To simplify the expression, we can combine the constants and rewrite the trigonometric functions using their identities. The identity sin(2x) = 2 sin(x) cos(x) can be used to rewrite the expression as dy/dx = -6 cos(x) sin(4x).

Final Answer:
Therefore, the derivative dy/dx of the given function y = 2 sin(3 cos 4x) is -6 cos(x) sin(4x).