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To solve the problem of finding the derivative of the function \( f(t) = -2t^2 - 7t - 2 \), we'll apply the basic rules of differentiation.

1. **Power Rule**: The derivative of \( t^n \) is \( n \cdot t^{n-1} \).
2. **Constant Multiple Rule**: The derivative of a constant times a function is the constant times the derivative of the function.
3. **Sum Rule**: The derivative of a sum of functions is the sum of their derivatives.

Now, let's differentiate each term in \( f(t) = -2t^2 - 7t - 2 \).

- The derivative of \( -2t^2 \) is \( -2 \cdot 2t^{2-1} = -4t \).
- The derivative of \( -7t \) is \( -7 \).
- The derivative of \( -2 \) (a constant) is \( 0 \).

Thus, the derivative of the function is:

\[
f'(t) = -4t - 7
\]

Would you like more details, or do you have any questions?

Here are 5 related questions to deepen your understanding:

1. What is the derivative of \( t^3 - 4t + 5 \)?
2. How do we apply the chain rule in differentiation?
3. Can we find the second derivative of \( f(t) = -2t^2 - 7t - 2 \)?
4. What are the common rules of differentiation beyond the power rule?
5. How can we apply differentiation in real-world problems?

**Tip:** When differentiating, always break the function down term by term and apply the appropriate rule for each term.

Use the rules of differentiation to find the derivative of the function f(t) = -2t^2 - 7t - 2.

Learn how to differentiate the polynomial function f(t) = -2t^2 - 7t - 2. The solution involves applying the power rule, sum rule, and constant rule. Suitable for high school students studying calculus.

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