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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.

Find dy/dx for the function y = x^(-3cos(x)).

Learn how to differentiate the function y = x^(-3cos(x)) using logarithmic differentiation and product rule. The step-by-step solution includes using implicit differentiation and trigonometric functions to solve this advanced calculus problem. Suitable for Grades 11-12.

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