To find the derivative of the function ( e^{4x} \cdot (2-3x) ), we can use the product rule.

Let's break down the function into two parts:
( f(x) = e^{4x} ) and ( g(x) = (2-3x) ).

The product rule states that if you have a product of two functions, the derivative of the product is the derivative of the first function times the second function plus the first function times the derivative of the second function.

Given ( f(x) = e^{4x} ) and ( g(x) = (2-3x) ), the derivative of the product is:

[ f'(x) \cdot g(x) + f(x) \cdot g'(x) ]

Let's find the derivatives separately:

( f'(x) = 4e^{4x} ) (using the chain rule for the derivative of ( e^{4x} ))

( g'(x) = -3 )

Now we can plug these values into our derivative formula:

[ f'(x) \cdot g(x) + f(x) \cdot g'(x) = (4e^{4x})(2-3x) + e^{4x}(-3) ]

Simplifying this expression we get:

[ 8e^{4x} - 12xe^{4x} - 3e^{4x} ]

So, the derivative of ( e^{4x} \cdot (2-3x) ) is ( 8e^{4x} - 12xe^{4x} - 3e^{4x} ).