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The given function is:

\[
f(u) = \frac{6}{\sqrt{u}}.
\]

To find the derivative, let's first rewrite the function in a more convenient form using exponent notation:

\[
f(u) = 6u^{-\frac{1}{2}}.
\]

Now, differentiate \( f(u) \) with respect to \( u \):

\[
f'(u) = \frac{d}{du} \left( 6u^{-\frac{1}{2}} \right).
\]

Using the power rule for differentiation, which states that \( \frac{d}{dx}(x^n) = nx^{n-1} \), we get:

\[
f'(u) = 6 \cdot \left( -\frac{1}{2} \right) u^{-\frac{1}{2} - 1}.
\]

Simplifying:

\[
f'(u) = -3 u^{-\frac{3}{2}}.
\]

Finally, we can express the result back in radical form:

\[
f'(u) = -\frac{3}{u^{\frac{3}{2}}} = -\frac{3}{u\sqrt{u}}.
\]

So, the derivative is:

\[
f'(u) = -\frac{3}{u\sqrt{u}}.
\]

Would you like more details on any step? 

Here are 5 related questions:
1. How is the power rule for differentiation applied in this context?
2. Can you explain how to rewrite functions involving radicals for easier differentiation?
3. What happens if the exponent is a positive fraction?
4. How would you integrate the function \(6u^{-\frac{1}{2}}\) instead of differentiating it?
5. How do higher-order derivatives behave for this type of function?

**Tip:** When differentiating functions with roots or fractional exponents, always rewrite the roots as powers for easier application of the power rule.

Find the derivative of the function by using the rules of differentiation: f(u) = 6 / sqrt(u)

Learn how to find the derivative of f(u) = 6 / sqrt(u) using the power rule in calculus. This step-by-step guide covers the rules of differentiation and simplification of functions involving radicals, suitable for high school students in Grades 11-12.

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