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

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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 $$ f(x) $$.
First derivative $$ f'(x) = 3 e^{x} (\cos(x) - \sin(x)) $$. Second derivative $$ f''(x) = -6 e^{x} \sin(x) $$.

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

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