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To find the derivative of the function \( f(x) = 24x^{\frac{3}{2}} - 6x^{\frac{1}{2}} \), we apply the power rule for differentiation, which states that if \( f(x) = ax^n \), then \( f'(x) = n \cdot ax^{n-1} \).

Let's differentiate each term:

1. For the term \( 24x^{\frac{3}{2}} \):
   \[
   f'(x) = \frac{3}{2} \cdot 24x^{\frac{3}{2} - 1} = 36x^{\frac{1}{2}}
   \]

2. For the term \( 6x^{\frac{1}{2}} \):
   \[
   f'(x) = \frac{1}{2} \cdot 6x^{\frac{1}{2} - 1} = 3x^{-\frac{1}{2}} = \frac{3}{\sqrt{x}}
   \]

Therefore, the derivative of the function \( f(x) \) is:
\[
f'(x) = 36x^{\frac{1}{2}} - \frac{3}{\sqrt{x}} = 36\sqrt{x} - \frac{3}{\sqrt{x}}
\]

Would you like further details or have any other questions? Here are some possible next steps:

1. Would you like to see how to simplify the expression \( 36\sqrt{x} - \frac{3}{\sqrt{x}} \)?
2. Are you interested in understanding how the power rule is derived?
3. Do you have any other functions you need to find the derivative for?
4. Would you like to learn about differentiating functions involving other fractional powers?
5. Do you want to know how to apply this derivative in practical problems?

**Tip**: When differentiating functions with fractional exponents, writing the exponents as fractions can help simplify the calculations and avoid common mistakes.

Find the derivative of the function f(x) = 24x^(3/2) - 6x^(1/2).

Learn how to find the derivative of the function f(x) = 24x^(3/2) - 6x^(1/2) using the power rule. This problem involves applying calculus principles to differentiate functions with fractional exponents. Suitable for advanced high school or college calculus students.

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