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The function provided is \( f(x) = \frac{12}{x^{\frac{3}{4}}} \).

To find the derivative, let's first rewrite the expression to make it easier:

\[
f(x) = 12x^{-\frac{3}{4}}
\]

Now, we can apply the power rule for differentiation, which is:

\[
\frac{d}{dx}\left(x^n\right) = n x^{n-1}
\]

Applying the power rule to \( f(x) = 12x^{-\frac{3}{4}} \):

\[
f'(x) = 12 \cdot \left(-\frac{3}{4}\right) x^{-\frac{3}{4} - 1} = -\frac{36}{4} x^{-\frac{7}{4}} = -9x^{-\frac{7}{4}}
\]

Thus, the derivative is:

\[
f'(x) = -9x^{-\frac{7}{4}}
\]

Would you like more details on this process or any clarifications? Here are some related questions you might find useful:

1. How can I apply the chain rule in similar cases?
2. What is the general rule for differentiating a power function?
3. How would the process change for a more complex numerator?
4. What are common mistakes to avoid in applying the power rule?
5. How can I integrate this function after finding the derivative?

**Tip:** When differentiating powers of \(x\), always simplify the expression first if possible to make applying the rules straightforward.

Look for the most convenient way to find the derivative. f(x) = 12 / x^(3/4).

Learn how to differentiate f(x) = 12 / x^(3/4) using the power rule. This problem covers key calculus concepts, including simplifying functions for differentiation and applying the power rule for solving derivatives. Suitable for students in advanced high school or first-year university calculus.

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