To find the derivative of the function $f(x) = \frac{3}{\sqrt{2x - 1}}$, we will use the chain rule and quotient rule.

Step-by-step solution:

Given: $f(x) = \frac{3}{\sqrt{2x - 1}}$ We can rewrite the square root in exponent form: $f(x) = 3(2x - 1)^{-\frac{1}{2}}$

Now, apply the chain rule:

1. Differentiate the outer function: $3(2x - 1)^{-\frac{1}{2}}$. The derivative of $(2x - 1)^{-\frac{1}{2}}$ is: $\frac{d}{dx} \left[ (2x - 1)^{-\frac{1}{2}} \right] = -\frac{1}{2}(2x - 1)^{-\frac{3}{2}} \cdot \frac{d}{dx}(2x - 1)$

2. Now, differentiate the inner function $2x - 1$: $\frac{d}{dx}(2x - 1) = 2$

3. Combining these, the derivative becomes: $f'(x) = 3 \cdot -\frac{1}{2}(2x - 1)^{-\frac{3}{2}} \cdot 2$ Simplifying: $f'(x) = -\frac{3}{(2x - 1)^{\frac{3}{2}}}$

Thus, the derivative of the function $f(x) = \frac{3}{\sqrt{2x - 1}}$ is: $f'(x) = -\frac{3}{(2x - 1)^{\frac{3}{2}}}$

Do you have any questions or would you like further clarification on this?

Here are 5 related questions:

1. How do you apply the chain rule for composite functions?
2. What is the power rule and how does it work for negative exponents?
3. How do you differentiate functions involving square roots?
4. Can you find the second derivative of $f(x)$?
5. How does the behavior of the derivative relate to the graph of the function?

Tip: When differentiating square roots, rewriting them as exponents often makes the differentiation process easier.