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We are tasked with finding the first, second, and third derivatives of the function:

\[
g(x) = -\frac{3}{8x}
\]

### Step 1: Simplify the given function
The function can be rewritten as:

\[
g(x) = -\frac{3}{8} \cdot x^{-1}
\]

Now, let's differentiate it.

### Step 2: First derivative \( g'(x) \)

We apply the power rule for differentiation:

\[
\frac{d}{dx}[x^n] = n \cdot x^{n-1}
\]

So,

\[
g'(x) = -\frac{3}{8} \cdot (-1) \cdot x^{-2}
\]

This simplifies to:

\[
g'(x) = \frac{3}{8x^2}
\]

This matches the result you entered, but it seems to have been marked wrong, possibly due to a formatting or interpretation issue in the system.

### Step 3: Second derivative \( g''(x) \)

To find the second derivative, differentiate \( g'(x) \):

\[
g''(x) = \frac{d}{dx} \left( \frac{3}{8x^2} \right)
\]

Applying the power rule again:

\[
g''(x) = \frac{3}{8} \cdot (-2) \cdot x^{-3} = -\frac{6}{8x^3} = -\frac{3}{4x^3}
\]

### Step 4: Third derivative \( g^{(3)}(x) \)

Now, let's differentiate \( g''(x) \):

\[
g^{(3)}(x) = \frac{d}{dx} \left( -\frac{3}{4x^3} \right)
\]

Again applying the power rule:

\[
g^{(3)}(x) = -\frac{3}{4} \cdot (-3) \cdot x^{-4} = \frac{9}{4x^4}
\]

### Final Results:
1. First derivative: \( g'(x) = \frac{3}{8x^2} \)
2. Second derivative: \( g''(x) = -\frac{3}{4x^3} \)
3. Third derivative: \( g^{(3)}(x) = \frac{9}{4x^4} \)

Would you like more details on any of these steps?

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Here are 5 related questions to further explore:

1. How do you apply the chain rule in differentiation problems?
2. What are some common applications of second and third derivatives in real-world problems?
3. Can you explain the difference between implicit and explicit differentiation?
4. What are the physical interpretations of higher-order derivatives?
5. How do you determine the critical points of a function using the first derivative?

**Tip:** Higher-order derivatives often give insights into the curvature and concavity of a graph, which are useful in optimization problems!

Find the first, second, and third derivative of the function g(x) = - (3 / 8x).

Learn how to find the first, second, and third derivatives of the function g(x) = -3/(8x) using the power rule. This problem involves higher-order derivatives, applying the power rule, and simplifying expressions. Suitable for students in Grades 11-12.

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