https://file.aiquickdraw.com/mathbot/file/1725993366198sblnwodj.jpg

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.

Given that y = x^3 - 10/x^4, where x ≠ 0, find dy/dx.

Learn how to differentiate the function y = x^3 - 10/x^4 step-by-step using the power rule. This problem is suitable for students in Grades 11-12 and covers key calculus concepts related to differentiation.

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