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The image provides the following mathematical problem:

**Problem Statement:**

Consider the function below:

\[
c(t) = t^2 - 1
\]

Find the first derivative (denoted as \( c'(t) \) or \(\frac{dc}{dt}\)) of the function.

**Solution:**

To find the first derivative of \( c(t) = t^2 - 1 \) with respect to \( t \), we can apply the power rule of differentiation.

The power rule states that if \( c(t) = t^n \), then \( c'(t) = n \cdot t^{n-1} \).

Applying this rule:

\[
\frac{dc}{dt} = \frac{d}{dt}(t^2 - 1)
\]

Differentiate each term separately:

1. The derivative of \( t^2 \) is \( 2t \).
2. The derivative of \( -1 \) is \( 0 \) (since the derivative of any constant is 0).

Therefore,

\[
\frac{dc}{dt} = 2t + 0 = 2t.
\]

**Answer:**

\[
\frac{dc}{dt} = 2t.
\]

Would you like a more detailed explanation, or do you have any questions?

Here are 5 related questions that expand on this topic:

1. How would you find the second derivative of \( c(t) = t^2 - 1 \)?
2. What is the geometric interpretation of the derivative \( \frac{dc}{dt} = 2t \)?
3. How does the derivative change if the function is \( c(t) = 3t^2 - 4t + 5 \)?
4. What are the critical points of the function \( c(t) = t^2 - 1 \)?
5. How do you apply the chain rule to differentiate a composite function?

**Tip:** Remember, the derivative represents the rate of change or the slope of the tangent line to the function at any given point.

Consider the function c(t) = t^2 - 1. Find the first derivative c'(t) = dc/dt.

Learn how to find the first derivative of the function c(t) = t^2 - 1 using the power rule of differentiation. This problem is ideal for students in Grades 11-12 who are studying calculus and the basics of differentiation.

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