To find $f' \left( \frac{\pi}{6} \right)$ for the function $f(x) = x e^x \cos(x)$, we'll follow these steps:

Step 1: Differentiate $f(x)$

The function is a product of three terms: $x$, $e^x$, and $\cos(x)$. To differentiate this, we will apply the product rule, which states:

$(fg)' = f'g + fg'$

For three functions $u(x), v(x), w(x)$, the product rule is extended as:

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

Here:

• $u(x) = x$
• $v(x) = e^x$
• $w(x) = \cos(x)$

Now, we calculate each derivative:

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

Using the product rule:

$f'(x) = (1)(e^x \cos(x)) + (x)(e^x \cos(x)) + (x)(e^x (-\sin(x)))$ $f'(x) = e^x \cos(x) + x e^x \cos(x) - x e^x \sin(x)$

Step 2: Substitute $x = \frac{\pi}{6}$

Now, substitute $x = \frac{\pi}{6}$ into the derivative:

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

We know the following trigonometric values:

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

Substitute these into the expression:

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

Step 3: Simplify and Calculate

Let's now calculate the values. I will compute the expression numerically.

The value of $f' \left( \frac{\pi}{6} \right)$ is approximately $1.785$, rounded to three decimal places.

Would you like further details or have any questions?

Here are some related questions to explore:

1. How is the product rule applied in differentiation?
2. How can we differentiate functions with more than two factors?
3. What is the chain rule, and how does it compare to the product rule?
4. How do you approximate the values of exponential and trigonometric functions?
5. What are some common applications of derivatives in physics?

Tip: When differentiating multiple products, keep track of each term carefully to avoid errors in the final expression!