We are given the function $f(x) = x e^x \cos(x)$ and asked to find the derivative, $f'(x)$, and then evaluate $f' \left( \frac{\pi}{6} \right)$.

Step 1: Differentiate $f(x)$

We will apply the product rule since $f(x)$ is a product of three functions: $x$, $e^x$, and $\cos(x)$. The product rule states:

$\frac{d}{dx} \left[ u(x)v(x)w(x) \right] = u'(x)v(x)w(x) + u(x)v'(x)w(x) + u(x)v(x)w'(x)$

Let $u(x) = x$, $v(x) = e^x$, and $w(x) = \cos(x)$. Now, find their derivatives:

• $u'(x) = 1$
• $v'(x) = e^x$
• $w'(x) = -\sin(x)$

Now, apply the product rule:

$f'(x) = u'(x)v(x)w(x) + u(x)v'(x)w(x) + u(x)v(x)w'(x)$ $f'(x) = 1 \cdot e^x \cdot \cos(x) + x \cdot e^x \cdot \cos(x) + x \cdot e^x \cdot (-\sin(x))$ $f'(x) = e^x \cos(x) + x e^x \cos(x) - x e^x \sin(x)$

Step 2: Evaluate $f' \left( \frac{\pi}{6} \right)$

Substitute $x = \frac{\pi}{6}$ into $f'(x)$:

$f' \left( \frac{\pi}{6} \right) = e^{\frac{\pi}{6}} \cos \left( \frac{\pi}{6} \right) + \frac{\pi}{6} e^{\frac{\pi}{6}} \cos \left( \frac{\pi}{6} \right) - \frac{\pi}{6} e^{\frac{\pi}{6}} \sin \left( \frac{\pi}{6} \right)$

We know:

• $\cos \left( \frac{\pi}{6} \right) = \frac{\sqrt{3}}{2}$
• $\sin \left( \frac{\pi}{6} \right) = \frac{1}{2}$

Now substitute these values:

$f' \left( \frac{\pi}{6} \right) = e^{\frac{\pi}{6}} \cdot \frac{\sqrt{3}}{2} + \frac{\pi}{6} e^{\frac{\pi}{6}} \cdot \frac{\sqrt{3}}{2} - \frac{\pi}{6} e^{\frac{\pi}{6}} \cdot \frac{1}{2}$

Step 3: Simplify and Calculate

Let me calculate the value of $f' \left( \frac{\pi}{6} \right)$ approximately.The value of $f' \left( \frac{\pi}{6} \right)$ is approximately 1.785.

Would you like more details on any part of the process? Here are 5 related questions you might want to explore:

1. How do we apply the product rule to functions with more than two terms?
2. Why is the exponential function $e^x$ so common in differentiation problems?
3. What are the steps for evaluating trigonometric values at specific angles?
4. How do we interpret the physical meaning of the derivative in different contexts?
5. Can we use alternative methods (e.g., chain rule) for solving such differentiation problems?

Tip: When working with product rules, it's useful to write down all derivative terms separately before applying the rule for better clarity.