find the derivative of $$ y=\sqrt[5]{x} $$

To find the derivative of the function

\[
y = \sqrt[5]{x}
\]

we can rewrite the function in exponent form:

\[
y = x^{1/5}
\]

### Step 1: Use the power rule

The **power rule** for derivatives states that if $ y = x^n $, then:

\[
\frac{dy}{dx} = n x^{n-1}
\]

Here, $ n = \frac{1}{5} $, so applying the power rule:

\[
\frac{dy}{dx} = \frac{1}{5} x^{\frac{1}{5} - 1}
\]

Simplify the exponent:

\[
\frac{dy}{dx} = \frac{1}{5} x^{-\frac{4}{5}}
\]

### Step 2: Final answer

Thus, the derivative of $ y = \sqrt[5]{x} $ is:

\[
\frac{dy}{dx} = \frac{1}{5} x^{-\frac{4}{5}}
\]

Alternatively, this can also be written as:

\[
\frac{dy}{dx} = \frac{1}{5} \cdot \frac{1}{x^{\frac{4}{5}}}
\]

Let me know if you'd like any clarifications!

Here are 5 related questions:
1. What is the derivative of $ y = \sqrt[3]{x} $?
2. How does the power rule apply if the function is $ y = x^n $ with negative exponents?
3. What is the second derivative of $ y = \sqrt[5]{x} $?
4. How does the chain rule apply if the function is $ y = \sqrt[5]{x^2} $?
5. How do you find the derivative of fractional powers like $ y = x^{3/5} $?

**Tip:** Always rewrite roots in exponent form to apply the power rule easily!

Learn how to find the derivative of y = \sqrt[5]{x} by rewriting the function in exponent form and applying the power rule. This problem demonstrates key calculus concepts and is ideal for students in Grades 11-12 and college-level calculus.

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