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The function given is \( f(x) = \frac{1}{2} \cdot x^{\frac{1}{8}} \). We are asked to find its derivative, \( f'(x) \).

To solve this:

1. The function is a product of a constant \( \frac{1}{2} \) and a power function \( x^{\frac{1}{8}} \).
2. The derivative of \( x^n \) with respect to \( x \) is \( n \cdot x^{n-1} \).

Let's apply the power rule to differentiate:

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

Simplifying:

\[
f'(x) = \frac{1}{16} \cdot x^{-\frac{7}{8}}
\]

Thus, the derivative is:

\[
f'(x) = \frac{1}{16x^{\frac{7}{8}}}
\]

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

Here are 5 related questions you might find helpful:
1. How do you differentiate more complex power functions?
2. What is the general rule for finding derivatives of products of constants and functions?
3. How would you integrate \( f'(x) = \frac{1}{16x^{7/8}} \) to find the original function?
4. What happens to the derivative if we modify the power in \( x^{1/8} \)?
5. How would you differentiate a similar function involving a negative exponent?

**Tip**: When differentiating a function with a constant multiplied by a variable term, keep the constant and just apply the derivative rules to the variable term.

Learn how to find the derivative of f(x) = (1/2) * x^(1/8) using the power rule. This step-by-step solution covers basic differentiation concepts and applies the power rule to solve for f'(x). Suitable for Grades 11-12 calculus students.

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