To find the derivative of \( y = x^{\frac{2}{7}} \) with respect to \( x \), we can use the power rule of differentiation, which states that if \( y = x^n \), then \( \frac{dy}{dx} = n \cdot x^{n-1} \).

Here's the step-by-step solution:

1. **Identify the exponent \( n \):**
   \[
   n = \frac{2}{7}
   \]

2. **Apply the power rule:**
   \[
   \frac{dy}{dx} = \frac{2}{7} \cdot x^{\frac{2}{7} - 1}
   \]

3. **Simplify the exponent:**
   \[
   \frac{2}{7} - 1 = \frac{2}{7} - \frac{7}{7} = -\frac{5}{7}
   \]

4. **Substitute and simplify:**
   \[
   \frac{dy}{dx} = \frac{2}{7} \cdot x^{-\frac{5}{7}}
   \]

So, the derivative is:

\[
\frac{dy}{dx} = \frac{2}{7} \cdot x^{-\frac{5}{7}}
\]

Would you like more details on any specific step or have any other questions?

Here are some related questions you might find interesting:

1. How do you find the second derivative of \( y = x^{\frac{2}{7}} \)?
2. What is the derivative of \( y = x^{n} \) for any real number \( n \)?
3. How do you differentiate functions involving fractional exponents?
4. Can you explain the power rule in differentiation in more detail?
5. How do you find the derivative of a composite function involving fractional exponents?

**Tip:** When dealing with negative exponents in differentiation, remember that \( x^{-n} = \frac{1}{x^n} \), which can be useful for simplifying expressions.

Find the derivative of y = x^(2/7) using the power rule of differentiation. This step-by-step solution covers important calculus concepts, including fractional exponents and applying the power rule. Ideal for high school students in Grades 11-12.

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