The given function is:

$y = x^{-3 \cos(x)}$

We need to find the derivative $\frac{dy}{dx}$. This is a product of a power function and a trigonometric function, so we will apply logarithmic differentiation.

Steps:

1. Take the natural logarithm of both sides of the equation to simplify the power: $\ln(y) = \ln(x^{-3 \cos(x)}) = -3 \cos(x) \ln(x)$

2. Differentiate both sides with respect to $x$. On the left side, use implicit differentiation, and on the right side, apply the product rule.

   For the left side: $\frac{d}{dx} [\ln(y)] = \frac{1}{y} \cdot \frac{dy}{dx}$

   For the right side: [ \frac{d}{dx} [-3 \cos(x) \ln(x)] = -3 \left( \frac{d}{dx} [\cos(x)] \ln(x) + \cos(x) \frac{d}{dx} [\ln(x)] \right) ] This expands to: $-3 \left( (-\sin(x)) \ln(x) + \cos(x) \cdot \frac{1}{x} \right)$ Simplifying further: $-3 \left( -\sin(x) \ln(x) + \frac{\cos(x)}{x} \right)$ This becomes: $3 \sin(x) \ln(x) - \frac{3 \cos(x)}{x}$

3. Combine both sides: $\frac{1}{y} \frac{dy}{dx} = 3 \sin(x) \ln(x) - \frac{3 \cos(x)}{x}$

4. Multiply by $y$ (which is $x^{-3 \cos(x)}$) to isolate $\frac{dy}{dx}$: $\frac{dy}{dx} = x^{-3 \cos(x)} \left( 3 \sin(x) \ln(x) - \frac{3 \cos(x)}{x} \right)$

Thus, the derivative of $y = x^{-3 \cos(x)}$ is:

$\frac{dy}{dx} = x^{-3 \cos(x)} \left( 3 \sin(x) \ln(x) - \frac{3 \cos(x)}{x} \right)$

Let me know if you need further clarification or have any questions! Here are some related questions for further practice:

1. How would you differentiate $y = x^{a \cos(x)}$ where $a$ is a constant?
2. Can you apply logarithmic differentiation to functions like $y = e^{x^2 \sin(x)}$?
3. How does the product rule work when differentiating products of functions?
4. What is the general form for the derivative of $x^n$ when $n$ is a function of $x$?
5. How do you differentiate a logarithmic function like $y = \ln(x^2 + x)$?

Tip: Always remember to apply the chain rule carefully when differentiating composite functions, especially when they involve products or powers.