To differentiate the function \( y = 4x^3 - x^4 \) with respect to \( x \), apply the power rule for differentiation, which states that for any term \( ax^n \), the derivative is \( anx^{n-1} \).

1. Differentiate \( 4x^3 \):
   \[
   \frac{d}{dx}(4x^3) = 4 \cdot 3x^{3-1} = 12x^2
   \]

2. Differentiate \( -x^4 \):
   \[
   \frac{d}{dx}(-x^4) = -4x^{4-1} = -4x^3
   \]

Combining these results, the derivative \( \frac{dy}{dx} \) is:
\[
\frac{dy}{dx} = 12x^2 - 4x^3
\]

### Final Answer:
\[
\frac{dy}{dx} = 12x^2 - 4x^3
\]

Would you like more details or have any questions?

### Related Questions
1. How do we apply the power rule for more complex functions?
2. What is the derivative of a function like \( y = x^4 - 3x^3 + x \)?
3. How can we find the second derivative of \( y = 4x^3 - x^4 \)?
4. What are the critical points of \( y = 4x^3 - x^4 \)?
5. How does differentiation relate to finding the slope of a curve?

#### Tip:
When differentiating terms individually, always watch for positive and negative signs to avoid common mistakes.

Discover the step-by-step process to differentiate the function y = 4x³ - x⁴ using the power rule of differentiation. Ideal for students in Grades 11-12, this solution explains each step and provides insights into key calculus concepts and differentiation techniques.

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