Find the second derivative of the function.
f(x)=e^x*cos(5x)

To find the second derivative of the function \( f(x) = e^x \cos(5x) \), we'll start by finding the first derivative and then differentiate again to get the second derivative.

### First Derivative

The function is a product of two functions, \( u(x) = e^x \) and \( v(x) = \cos(5x) \). We'll use the product rule, which states:

\[
(fg)' = f'g + fg'
\]

So, the first derivative \( f'(x) \) is:

\[
f'(x) = \frac{d}{dx}(e^x) \cdot \cos(5x) + e^x \cdot \frac{d}{dx}(\cos(5x))
\]

Now, compute the derivatives:

\[
\frac{d}{dx}(e^x) = e^x
\]
\[
\frac{d}{dx}(\cos(5x)) = -5\sin(5x)
\]

Substitute these into the product rule:

\[
f'(x) = e^x \cdot \cos(5x) - 5e^x \cdot \sin(5x)
\]

Factor out the common factor \( e^x \):

\[
f'(x) = e^x (\cos(5x) - 5\sin(5x))
\]

### Second Derivative

Now, to find the second derivative \( f''(x) \), we differentiate \( f'(x) = e^x (\cos(5x) - 5\sin(5x)) \) again using the product rule:

\[
f''(x) = \frac{d}{dx}(e^x) \cdot (\cos(5x) - 5\sin(5x)) + e^x \cdot \frac{d}{dx}(\cos(5x) - 5\sin(5x))
\]

We already know that:

\[
\frac{d}{dx}(e^x) = e^x
\]

Now, differentiate \( \cos(5x) - 5\sin(5x) \):

\[
\frac{d}{dx}(\cos(5x)) = -5\sin(5x)
\]
\[
\frac{d}{dx}(-5\sin(5x)) = -25\cos(5x)
\]

So:

\[
\frac{d}{dx}(\cos(5x) - 5\sin(5x)) = -5\sin(5x) - 25\cos(5x)
\]

Substitute back into the product rule:

\[
f''(x) = e^x (\cos(5x) - 5\sin(5x)) + e^x (-5\sin(5x) - 25\cos(5x))
\]

Factor out \( e^x \) again:

\[
f''(x) = e^x [(\cos(5x) - 5\sin(5x)) + (-5\sin(5x) - 25\cos(5x))]
\]

Simplify the expression inside the brackets:

\[
f''(x) = e^x (\cos(5x) - 25\cos(5x) - 5\sin(5x) - 5\sin(5x))
\]

Combine like terms:

\[
f''(x) = e^x (-24\cos(5x) - 10\sin(5x))
\]

So the second derivative of the function \( f(x) = e^x \cos(5x) \) is:

\[
f''(x) = e^x (-24\cos(5x) - 10\sin(5x))
\]

Would you like more details or have any questions?

Here are some related questions:
1. How would you compute the third derivative of \( f(x) \)?
2. How does the second derivative inform us about the concavity of \( f(x) \)?
3. What are the critical points of the second derivative?
4. How do exponential and trigonometric functions combine in higher derivatives?
5. Can the result be interpreted in terms of oscillations and exponential growth?

**Tip:** When differentiating complex functions, always simplify each derivative before moving to the next to reduce calculation errors.

Learn how to find the second derivative of the function f(x) = e^x * cos(5x) step-by-step. This problem involves differentiating exponential and trigonometric functions using the product rule. Suitable for advanced high school students and calculus enthusiasts.

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