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To solve the problem of finding the first and second derivatives of the function \( y = -2x^4 - 1 \), we will apply the basic rules of differentiation.

### Step 1: First Derivative
The original function is:
\[
y = -2x^4 - 1
\]

Using the power rule \( \frac{d}{dx} \left( x^n \right) = nx^{n-1} \), let's differentiate each term with respect to \(x\).

- The derivative of \( -2x^4 \) is:
  \[
  \frac{d}{dx} (-2x^4) = -2 \cdot 4x^{4-1} = -8x^3
  \]

- The derivative of the constant term \( -1 \) is 0, as the derivative of any constant is 0.

Thus, the first derivative \( \frac{dy}{dx} \) is:
\[
\frac{dy}{dx} = -8x^3
\]

### Step 2: Second Derivative
Now, let's differentiate \( -8x^3 \) to find the second derivative.

- The derivative of \( -8x^3 \) is:
  \[
  \frac{d}{dx} (-8x^3) = -8 \cdot 3x^{3-1} = -24x^2
  \]

Thus, the second derivative \( \frac{d^2y}{dx^2} \) is:
\[
\frac{d^2y}{dx^2} = -24x^2
\]

### Final Answer:
- First derivative: \( \frac{dy}{dx} = -8x^3 \)
- Second derivative: \( \frac{d^2y}{dx^2} = -24x^2 \)

Would you like further details or explanations on any step?

Here are 5 related questions to deepen your understanding:
1. What is the power rule for derivatives, and how does it apply to different functions?
2. How would the derivatives change if the function had a higher degree polynomial?
3. What is the meaning of the first derivative in the context of a graph?
4. How does the second derivative relate to the concavity of the function's graph?
5. What are higher-order derivatives, and when are they used?

**Tip:** Remember, the first derivative gives the slope (rate of change) of the function, while the second derivative provides information about the curvature (concavity) of the function.

Find the first and second derivatives of y = -2x^4 - 1.

Learn how to find the first and second derivatives of the function y = -2x^4 - 1 using the power rule of differentiation. This problem involves key calculus concepts and is suitable for Grade 12 or undergraduate students.

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