https://file.aiquickdraw.com/mathbot/file/17271178692854zbvaux5.jpg

The problem in the image asks for the derivative of the function \( f(x) = x^{1/5} \) at \( x = 0.6 \). Let's go step by step to calculate the derivative and check the given result.

### Step 1: Differentiate \( f(x) = x^{1/5} \)
Using the power rule for differentiation, \( f'(x) = \frac{d}{dx} \left( x^n \right) = n \cdot x^{n-1} \), where \( n = \frac{1}{5} \):

\[
f'(x) = \frac{1}{5} \cdot x^{1/5 - 1} = \frac{1}{5} \cdot x^{-4/5}
\]

### Step 2: Evaluate \( f'(x) \) at \( x = 0.6 \)

Now substitute \( x = 0.6 \) into the derivative:

\[
f'(0.6) = \frac{1}{5} \cdot (0.6)^{-4/5}
\]

I will compute the exact value of this expression.The calculated value of \( f'(0.6) \) is approximately \( 0.30096 \), which matches the given exact value \( 0.300960150482478 \) in the image.

Would you like more details on the steps, or do you have any questions?

Here are some related questions:
1. How do you apply the power rule for differentiation in general?
2. What are the rules for differentiating functions involving fractional exponents?
3. How does negative exponents in derivatives affect the result?
4. Can you explain how to differentiate a composite function?
5. What is the significance of evaluating derivatives at specific points?

**Tip:** When differentiating functions with fractional exponents, always simplify the exponent rules first to make calculations easier.

Differentiate the function f(x) = x^(1/5) and find the derivative at x = 0.6

Learn how to differentiate the function f(x) = x^(1/5) using the power rule and evaluate the derivative at x = 0.6. This solution demonstrates step-by-step differentiation for functions with fractional exponents. Suitable for calculus students in Grades 11-12 or early university courses.

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