The equation in the image is:

$y = x^3 - \frac{5}{x^3} - \frac{10}{x^4}$

You are asked to find the derivative $\frac{dy}{dx}$.

Let’s proceed with finding the derivative step-by-step:

Given function:

$y = x^3 - \frac{5}{x^3} - \frac{10}{x^4}$

Step 1: Rewrite in terms of powers of $x$

$y = x^3 - 5x^{-3} - 10x^{-4}$

Step 2: Apply the power rule to differentiate

The power rule states that $\frac{d}{dx} \left( x^n \right) = nx^{n-1}$.

$\frac{dy}{dx} = \frac{d}{dx}(x^3) - \frac{d}{dx}(5x^{-3}) - \frac{d}{dx}(10x^{-4})$

Step 3: Differentiate term-by-term

• $\frac{d}{dx}(x^3) = 3x^2$
• $\frac{d}{dx}(5x^{-3}) = -15x^{-4}$
• $\frac{d}{dx}(10x^{-4}) = -40x^{-5}$

Step 4: Combine results

$\frac{dy}{dx} = 3x^2 + 15x^{-4} + 40x^{-5}$

This is the derivative of the given function.

Would you like any further clarifications or details?

Here are 5 related questions you might want to explore:

1. What is the second derivative of the given function?
2. How would the result change if a constant term was added to $y$?
3. Can you evaluate $\frac{dy}{dx}$ at specific values of $x$?
4. How can implicit differentiation be used in more complex scenarios?
5. What is the geometric interpretation of the derivative of a function?

Tip: When differentiating, remember that negative exponents are treated like regular powers, and differentiation follows the same power rule.