https://file.aiquickdraw.com/mathbot/file/1725690454063negmzh8c.jpg

Let's analyze the image.

The given function is:

\[
h(t) = 4t^{3/2} - 12t^{-3/2}
\]

The questions are:

1. Find \( h'(t) \) (the first derivative of \( h(t) \)).
2. Evaluate \( h'(5) \).
3. Find \( h''(t) \) (the second derivative of \( h(t) \)).
4. Evaluate \( h''(5) \).

### Step 1: Find the first derivative \( h'(t) \)

We differentiate \( h(t) \) with respect to \( t \).

\[
h'(t) = \frac{d}{dt} \left( 4t^{3/2} - 12t^{-3/2} \right)
\]

Using the power rule \( \frac{d}{dt}(t^n) = n t^{n-1} \), let's differentiate each term.

### Step 2: Evaluate \( h'(5) \)

Substitute \( t = 5 \) into the first derivative.

### Step 3: Find the second derivative \( h''(t) \)

Differentiate \( h'(t) \) with respect to \( t \) to find the second derivative \( h''(t) \).

### Step 4: Evaluate \( h''(5) \)

Substitute \( t = 5 \) into the second derivative.

I'll proceed with these calculations now.Here are the results:

1. The first derivative \( h'(t) \) is:
   \[
   h'(t) = 6t^{1/2} + 18t^{-5/2}
   \]

2. The value of \( h'(5) \) is approximately:
   \[
   h'(5) \approx 13.738
   \]

3. The second derivative \( h''(t) \) is:
   \[
   h''(t) = 3t^{-1/2} - 45t^{-7/2}
   \]

4. The value of \( h''(5) \) is approximately:
   \[
   h''(5) \approx 1.181
   \]

Would you like further details or explanations on these steps?

Here are five related questions to further explore:

1. What is the significance of the first derivative \( h'(t) \)?
2. How can we interpret the second derivative \( h''(t) \) in terms of concavity?
3. What does the value of \( h'(5) \) represent in this context?
4. How would the third derivative \( h'''(t) \) be calculated?
5. What applications could this type of function have in physics or engineering?

**Tip:** When differentiating powers of \( t \), always remember the power rule and to adjust the exponent accordingly after differentiation!

Let h(t) = 4t^(3/2) - 12t^(-3/2). Then find h'(t), evaluate h'(5), find h''(t), and evaluate h''(5).

Learn how to find the first and second derivatives of the function h(t) = 4t^(3/2) - 12t^(-3/2), and how to evaluate these derivatives at specific points. This problem covers key calculus concepts, including the power rule and derivative rules. Suitable for students in Grades 11-12 or College-level Calculus.

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