We are given the function $f(x) = 3e^{x} \cos(x)$ and tasked with finding its first and second derivatives, $f'(x)$ and $f''(x)$.

Step 1: First Derivative $f'(x)$

We will use the product rule to differentiate this function. The product rule states:

$(f \cdot g)' = f' \cdot g + f \cdot g'$

Let:

• $u(x) = 3e^x$
• $v(x) = \cos(x)$

Now, find the derivatives of $u(x)$ and $v(x)$:

$u'(x) = 3e^x$ $v'(x) = -\sin(x)$

Now, apply the product rule:

$f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)$ $f'(x) = (3e^x) \cdot \cos(x) + (3e^x) \cdot (-\sin(x))$ $f'(x) = 3e^x \cos(x) - 3e^x \sin(x)$

Thus, the first derivative is:

$f'(x) = 3e^x (\cos(x) - \sin(x))$

Step 2: Second Derivative $f''(x)$

Next, we need to differentiate $f'(x)$ again. We apply the product rule to each term in $f'(x) = 3e^x (\cos(x) - \sin(x))$.

Let:

• $u(x) = 3e^x$
• $v(x) = \cos(x) - \sin(x)$

First, differentiate each part: $u'(x) = 3e^x$ $v'(x) = -\sin(x) - \cos(x)$

Now apply the product rule again:

$f''(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)$ $f''(x) = (3e^x) \cdot (\cos(x) - \sin(x)) + (3e^x) \cdot (-\sin(x) - \cos(x))$ $f''(x) = 3e^x (\cos(x) - \sin(x)) + 3e^x (-\sin(x) - \cos(x))$

Simplify the expression:

$f''(x) = 3e^x [(\cos(x) - \sin(x)) + (-\sin(x) - \cos(x))]$ $f''(x) = 3e^x (0 - 2\sin(x))$ $f''(x) = -6e^x \sin(x)$

Thus, the second derivative is:

$f''(x) = -6e^x \sin(x)$

Final Results:

$f'(x) = 3e^x (\cos(x) - \sin(x))$ $f''(x) = -6e^x \sin(x)$

Would you like further details or have any questions?

Here are 5 related questions you might find helpful:

1. How do you apply the product rule for functions involving exponentials and trigonometric terms?
2. What is the physical or geometric interpretation of the first derivative in different scenarios?
3. How does the chain rule differ from the product rule in differentiation?
4. What are some common errors to avoid when applying the product rule?
5. Can we further analyze the behavior of $f(x)$ at specific points like $x = 0$?

Tip: The product rule is essential for handling derivatives of functions where terms are multiplied, especially when combining exponential and trigonometric functions.