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:

$\frac{dy}{dx} = 12x^2 - 4x^3$

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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.