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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.

Differentiate the function g(x) = 1 / sqrt(x) + sqrt[5](x)

Learn how to differentiate the function g(x) = 1 / sqrt(x) + sqrt[5](x). This solution walks through using the power rule and exponent rules to compute the derivative of the function. Suitable for students in Grades 11-12.

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