To differentiate the function $y = x + \frac{5}{x^3} + x - 1$, we'll apply the rules of differentiation to each term.

Step-by-Step:

1. Differentiate $x$: $\frac{d}{dx}(x) = 1$

2. Differentiate $\frac{5}{x^3}$: Rewrite the term as $5x^{-3}$, then differentiate using the power rule: $\frac{d}{dx}(5x^{-3}) = 5 \cdot (-3x^{-4}) = -\frac{15}{x^4}$

3. Differentiate $x$ (again): $\frac{d}{dx}(x) = 1$

4. Differentiate $-1$: $\frac{d}{dx}(-1) = 0$

Putting it all together:

The derivative of the function is: $y' = 1 - \frac{15}{x^4} + 1 + 0$ Simplifying: $y' = 2 - \frac{15}{x^4}$

So the derivative of the function is: $y' = 2 - \frac{15}{x^4}$

Would you like further details or clarifications on any part?

Here are some related questions you might find interesting:

1. How would you differentiate higher-order terms, such as $\frac{5}{x^n}$?
2. How do you apply the power rule for negative exponents?
3. What are the common mistakes people make when differentiating rational functions?
4. How would you handle differentiating a product of two functions?
5. What happens when you differentiate an exponential function?

Tip: Always rewrite fractions as powers of $x$ before differentiating, especially when dealing with terms like $\frac{1}{x^n}$. This makes applying the power rule much simpler.