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The function given is \( f(x) = x^6 e^x \cos(x) \), and we are tasked with finding its derivative \( f'(x) \).

This expression involves a product of three functions: 
- \( x^6 \),
- \( e^x \),
- \( \cos(x) \).

We will use the **product rule** for derivatives, which states that for functions \( u(x) \), \( v(x) \), and \( w(x) \):

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

For three functions, this becomes:
\[
(u \cdot v \cdot w)' = u' \cdot v \cdot w + u \cdot v' \cdot w + u \cdot v \cdot w'.
\]

Let:
- \( u(x) = x^6 \),
- \( v(x) = e^x \),
- \( w(x) = \cos(x) \).

Now, we compute the individual derivatives:
1. \( u'(x) = \frac{d}{dx}(x^6) = 6x^5 \),
2. \( v'(x) = \frac{d}{dx}(e^x) = e^x \),
3. \( w'(x) = \frac{d}{dx}(\cos(x)) = -\sin(x) \).

Now applying the product rule:

\[
f'(x) = u'(x) \cdot v(x) \cdot w(x) + u(x) \cdot v'(x) \cdot w(x) + u(x) \cdot v(x) \cdot w'(x)
\]

Substituting the values:
\[
f'(x) = 6x^5 \cdot e^x \cdot \cos(x) + x^6 \cdot e^x \cdot \cos(x) + x^6 \cdot e^x \cdot (-\sin(x))
\]

Simplify:
\[
f'(x) = e^x \left( 6x^5 \cos(x) + x^6 \cos(x) - x^6 \sin(x) \right)
\]

Thus, the derivative is:

\[
f'(x) = e^x \left( (6x^5 + x^6) \cos(x) - x^6 \sin(x) \right).
\]

Would you like further details or clarifications?

Here are 5 related questions:
1. How would the derivative change if the function involved sine instead of cosine?
2. What happens if we add another function, say \( \ln(x) \), to the original function?
3. Can you apply the product rule to more than three functions? How?
4. What is the second derivative of the function \( f(x) = x^6 e^x \cos(x) \)?
5. How does the behavior of the derivative change as \( x \) approaches infinity?

**Tip**: Always look for ways to simplify expressions after applying the product rule; often terms can be factored for easier handling!

Find the derivative of f(x) = x^6 e^x cos(x).

This page provides a step-by-step guide to finding the derivative of the function f(x) = x^6 e^x cos(x) using the product rule. It involves the differentiation of powers, exponentials, and trigonometric functions, suitable for college-level calculus.

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