if the deritive of f(x) = lim as h goes to zero of (the cube root of (8+6h+h^2)-2)/h what is f(x)

To find the function \( f(x) \) given its derivative in the form of a limit, we start with the expression:

\[
f'(x) = \lim_{h \to 0} \frac{\sqrt[3]{8 + 6h + h^2} - 2}{h}
\]

This expression resembles the definition of the derivative using limits, suggesting it might be the derivative of a function evaluated at a specific point. Let's evaluate this limit step-by-step to identify the function \( f(x) \).

### Step 1: Recognize the Expression

The expression inside the limit resembles the form of the derivative of the cube root function, \( \sqrt[3]{x} \). More specifically, it looks like the derivative of \( f(x) = \sqrt[3]{x} \) at some point.

### Step 2: Identify the Point

The expression \(\sqrt[3]{8 + 6h + h^2}\) suggests that it is centered around the point where \( x = 8 \). This is because when \( h = 0 \), the cube root expression becomes \(\sqrt[3]{8} = 2\).

### Step 3: Set up the General Derivative Formula

The derivative of the cube root function, \( f(x) = \sqrt[3]{x} \), is:

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

Let's evaluate this derivative at \( x = 8 \):

\[
f'(8) = \frac{1}{3} \times 8^{-\frac{2}{3}} = \frac{1}{3} \times \frac{1}{4} = \frac{1}{12}
\]

### Step 4: Evaluate the Limit

We have previously identified that the given limit evaluates the derivative at the point \( x = 8 \), and the limit should match this evaluation:

\[
\lim_{h \to 0} \frac{\sqrt[3]{8 + 6h + h^2} - 2}{h} = \frac{1}{12}
\]

Thus, the function \( f(x) \) whose derivative gives the limit expression is:

\[
f(x) = \sqrt[3]{x}
\]

The derivative calculation above confirms that this matches the given limit, as the derivative of \( \sqrt[3]{x} \) evaluated at \( x = 8 \) gives \(\frac{1}{12}\).

### Conclusion

Therefore, the function \( f(x) \) is:

\[
f(x) = \sqrt[3]{x}
\]

Would you like to see more details, or do you have any questions about this explanation?

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Here are some additional related questions you might find interesting:

1. What is the derivative of \( f(x) = \sqrt[3]{x} \) at \( x = 1 \)?
2. How do you differentiate \( f(x) = x^{1/3} \) using the power rule?
3. Can you find the derivative of \( f(x) = \sqrt[3]{x^3 + 1} \)?
4. What is the derivative of \( g(x) = \sqrt[3]{x^2} \)?
5. How can you approximate \( \sqrt[3]{9} \) using the derivative of \( f(x) = \sqrt[3]{x} \)?
6. What are the critical points of \( f(x) = \sqrt[3]{x} \)?
7. How does the graph of \( f(x) = \sqrt[3]{x} \) compare to \( g(x) = x^{1/3} \)?
8. How does the derivative of \( f(x) = \sqrt[3]{x} \) help in understanding its concavity and inflection points?

**Tip:** To differentiate functions involving roots, consider rewriting them as powers, such as \( x^{1/3} \), to apply the power rule easily.

Explore how to find the function f(x) from its derivative limit \( \lim_{h \to 0} \frac{\sqrt[3]{8 + 6h + h^2} - 2}{h} \). Understand the definition of the derivative using limits and apply it to identify f(x). Detailed solution with mathematical formulas and step-by-step evaluation.

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