The function given is:

$g(x) = \frac{1}{\sqrt{x}} + \sqrt[5]{x}$

We will differentiate this function step by step.

Step 1: Rewrite the terms

$g(x) = x^{-\frac{1}{2}} + x^{\frac{1}{5}}$

Step 2: Differentiate each term

1. Differentiate $x^{-\frac{1}{2}}$: $\frac{d}{dx}(x^{-\frac{1}{2}}) = -\frac{1}{2} x^{-\frac{3}{2}}$

2. Differentiate $x^{\frac{1}{5}}$: $\frac{d}{dx}(x^{\frac{1}{5}}) = \frac{1}{5} x^{-\frac{4}{5}}$

Step 3: Combine the results

$g'(x) = -\frac{1}{2} x^{-\frac{3}{2}} + \frac{1}{5} x^{-\frac{4}{5}}$

So, the derivative of the function is:

$g'(x) = -\frac{1}{2} x^{-\frac{3}{2}} + \frac{1}{5} x^{-\frac{4}{5}}$

Would you like further clarification or details on any step?

Here are 5 related questions to expand your understanding:

1. How would you find the second derivative of $g(x)$?
2. What does the negative exponent signify in terms of the graph of the function?
3. How would you apply the chain rule if the exponents were functions of $x$?
4. How do we handle more complex fractional exponents in derivatives?
5. How would this function behave as $x \to 0$ and $x \to \infty$?

Tip: Always rewrite roots and fractions in terms of exponents for easier differentiation.