A graphing calculator is recommended. Let \( f(x)=3 e^{x} \cos (x) \) (a) Find \( f^{\prime}(x) \) and \( f^{\prime \prime}(x) \)

To find the first and second derivatives of the function \( f(x) = 3 e^{x} \cos(x) \), we'll use the product rule and the chain rule from calculus. ### (a) First Derivative \( f'(x) \) **Function:** \[ f(x) = 3 e^{x} \cos(x) \] **Applying the Product Rule:** \[ f'(x) = 3 \left( \frac{d}{dx} \left( e^{x} \cos(x) \right) \right) \] \[ f'(x) = 3 \left( e^{x} \cos(x) + e^{x} (-\sin(x)) \right) \] \[ f'(x) = 3 e^{x} (\cos(x) - \sin(x)) \] **Simplified First Derivative:** \[ f'(x) = 3 e^{x} (\cos(x) - \sin(x)) \] ### Second Derivative \( f''(x) \) **First Derivative:** \[ f'(x) = 3 e^{x} (\cos(x) - \sin(x)) \] **Applying the Product Rule Again:** \[ f''(x) = 3 \left( \frac{d}{dx} \left( e^{x} (\cos(x) - \sin(x)) \right) \right) \] \[ f''(x) = 3 \left( e^{x} (\cos(x) - \sin(x)) + e^{x} (-\sin(x) - \cos(x)) \right) \] \[ f''(x) = 3 e^{x} (\cos(x) - \sin(x) - \sin(x) - \cos(x)) \] \[ f''(x) = 3 e^{x} (-2 \sin(x)) \] \[ f''(x) = -6 e^{x} \sin(x) \] **Simplified Second Derivative:** \[ f''(x) = -6 e^{x} \sin(x) \] ### Summary - **First Derivative:** \[ f'(x) = 3 e^{x} (\cos(x) - \sin(x)) \] - **Second Derivative:** \[ f''(x) = -6 e^{x} \sin(x) \] These derivatives can be used for further analysis, such as finding critical points, concavity, and points of inflection for the function \( f(x) \).