The problem is to find the derivative of the function $f(x) = \cos(x) \cdot e^x$.

1. Problem Description

The problem is to find the derivative of the function $f(x) = \cos(x) \cdot e^x$.

2. Solution Steps

To find the derivative of $f(x) = \cos(x) \cdot e^x$, we will use the product rule. The product rule states that if $f(x) = u(x) \cdot v(x)$, then $f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)$.

In this case, let $u(x) = \cos(x)$ and $v(x) = e^x$.

Then, $u'(x) = -\sin(x)$ and $v'(x) = e^x$.

Using the product rule, we have:

$f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)$

$f'(x) = (-\sin(x)) \cdot e^x + \cos(x) \cdot e^x$

$f'(x) = e^x(\cos(x) - \sin(x))$

3. Final Answer

The derivative of the function is $e^x(\cos(x) - \sin(x))$.