To differentiate the function $y = \frac{1}{5x^5}$, we can rewrite it in a more convenient form for differentiation.

First, express the function as:

$y = \frac{1}{5} x^{-5}$

Now, differentiate this with respect to $x$, using the power rule $\frac{d}{dx} \left( x^n \right) = n x^{n-1}$:

$\frac{dy}{dx} = \frac{1}{5} \cdot (-5) x^{-5-1} = -\frac{5}{5} x^{-6}$

Simplifying this:

$\frac{dy}{dx} = -x^{-6} = -\frac{1}{x^6}$

Thus, the derivative of $y = \frac{1}{5x^5}$ is:

$\frac{dy}{dx} = -\frac{1}{x^6}$

Let me know if you'd like further details or explanations!

Here are some related questions for practice:

1. Differentiate $y = 3x^4 - 2x^2 + 5x$.
2. Differentiate $y = \frac{1}{x^3}$.
3. Differentiate $y = 7x^{-2} + 4x^5$.
4. Find the derivative of $y = \frac{2}{x^6}$.
5. Differentiate $y = x^7 + 2x^3 - 5x$.

Tip: Remember that when differentiating a term like $x^{-n}$, use the power rule and subtract 1 from the exponent to get the new power.