To find the derivative of the given function $y = x^{4 \cos(x)}$ using logarithmic differentiation, we can follow these steps:

Step 1: Take the natural logarithm of both sides.

We begin by applying the natural logarithm to both sides of the equation to simplify the power:

$\ln(y) = \ln\left(x^{4 \cos(x)}\right)$

Using the property of logarithms that $\ln(a^b) = b \ln(a)$, we simplify:

$\ln(y) = 4 \cos(x) \cdot \ln(x)$

Step 2: Differentiate both sides with respect to $x$.

Now, we differentiate both sides of the equation. On the left-hand side, use the chain rule:

$\frac{d}{dx} [\ln(y)] = \frac{1}{y} \cdot \frac{dy}{dx}$

On the right-hand side, we apply the product rule to $4 \cos(x) \cdot \ln(x)$:

$\frac{d}{dx} [4 \cos(x) \ln(x)] = 4 \left( -\sin(x) \ln(x) + \frac{\cos(x)}{x} \right)$

Step 3: Solve for $\frac{dy}{dx}$.

Now, we equate the derivatives:

$\frac{1}{y} \cdot \frac{dy}{dx} = 4 \left( -\sin(x) \ln(x) + \frac{\cos(x)}{x} \right)$

To solve for $\frac{dy}{dx}$, multiply both sides by $y$ (which is $y = x^{4 \cos(x)}$):

$\frac{dy}{dx} = x^{4 \cos(x)} \cdot 4 \left( -\sin(x) \ln(x) + \frac{\cos(x)}{x} \right)$

Thus, the derivative of $y = x^{4 \cos(x)}$ using logarithmic differentiation is:

$\frac{dy}{dx} = x^{4 \cos(x)} \cdot 4 \left( -\sin(x) \ln(x) + \frac{\cos(x)}{x} \right)$

Would you like further clarification or a deeper breakdown of any step?

Here are some related questions for further practice:

1. How does the product rule apply in logarithmic differentiation?
2. What is the importance of the chain rule in logarithmic differentiation?
3. Can you apply logarithmic differentiation to any type of function? Why or why not?
4. How would you differentiate $y = e^{x \sin(x)}$ using logarithmic differentiation?
5. How do logarithmic properties simplify complex exponents in differentiation?

Tip: When using logarithmic differentiation, always apply the chain rule carefully, especially if the function involves both powers and trigonometric terms!