To differentiate $f(x) = 9e^{x \cos x}$, we will apply the chain rule.

Steps:

1. Let $u(x) = x \cos x$. First, differentiate $u(x)$ using the product rule: $u'(x) = \frac{d}{dx} \left( x \cos x \right) = \cos x - x \sin x$

2. Differentiate $f(x)$. Now, apply the chain rule to differentiate the exponential function: $f'(x) = 9 \cdot e^{x \cos x} \cdot \frac{d}{dx} \left( x \cos x \right)$ Using the result from step 1: $f'(x) = 9e^{x \cos x} \cdot \left( \cos x - x \sin x \right)$

So, the derivative is: $f'(x) = 9e^{x \cos x} \left( \cos x - x \sin x \right)$

Would you like a detailed breakdown of the differentiation process or further explanation?

Related Questions:

1. What is the product rule and when is it used?
2. How do you apply the chain rule in differentiation?
3. How would you differentiate other functions like $x e^{x \sin x}$?
4. What is the derivative of trigonometric functions like $\cos x$ and $\sin x$?
5. How does the derivative of exponential functions change with different exponents?

Tip:

When using the chain rule, always identify the inner and outer functions first, which will guide you through the differentiation steps.