Given:
f(x) = 8e*cos^2(x)

To differentiate f(x) with respect to x, we can use the chain rule and the product rule.

Step 1: Apply the chain rule
The chain rule states that if we have a composition of functions, then the derivative of the composition is the derivative of the outer function multiplied by the derivative of the inner function.

In this case, the outer function is e^x and the inner function is cos^2(x).

The derivative of e^x with respect to x is e^x.
The derivative of cos^2(x) with respect to x can be found using the chain rule as follows:
Let u = cos(x)
Then, cos^2(x) = u^2
Taking the derivative of cos^2(x) with respect to x using the chain rule:
d/dx (cos^2(x)) = d/dx (u^2) = 2u * du/dx

So, d/dx (cos^2(x)) = 2cos(x) * (-sin(x)) = -2sin(x)cos(x)

Therefore, the derivative of the composition e*cos^2(x) with respect to x is:
d/dx (e*cos^2(x)) = e^x * (-2sin(x)cos(x))

Step 2: Simplify the expression
We can simplify the expression -2sin(x)cos(x) using the identity sin(2x) = 2sin(x)cos(x).

Therefore, -2sin(x)cos(x) = -sin(2x)

Step 3: Multiply by the constant
The original function is f(x) = 8e*cos^2(x)

So, the derivative of f(x) with respect to x is:
d/dx (f(x)) = 8e * d/dx (cos^2(x))

Substituting the earlier expression for d/dx (cos^2(x)):
d/dx (f(x)) = 8e * (-sin(2x))

Step 4: Final result
The final result is -16e*sin(2x)

Therefore, the correct answer is option B, -16 sin(2x) e*cos^2(x).