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!