We are asked to find the derivative of the function:

$h(x) = \left( 5x - 3 - 5x + 8 \right)^{\frac{1}{3}} - 9x^{-3} + 6x^4$

Step 1: Simplify the function

We can simplify the terms inside the first expression:

$5x - 3 - 5x + 8 = 5$

Thus, the function simplifies to:

$h(x) = 5^{\frac{1}{3}} - 9x^{-3} + 6x^4$

Since $5^{\frac{1}{3}}$ is a constant, its derivative will be zero.

Step 2: Differentiate the remaining terms

Now we differentiate term by term:

1. $\frac{d}{dx}\left( 5^{\frac{1}{3}} \right) = 0$ because the derivative of a constant is zero.
2. $\frac{d}{dx}\left( -9x^{-3} \right) = -9 \cdot (-3) x^{-3-1} = 27x^{-4}$
3. $\frac{d}{dx}\left( 6x^4 \right) = 6 \cdot 4x^{4-1} = 24x^3$

Step 3: Combine the results

The derivative of $h(x)$ is:

$h'(x) = 27x^{-4} + 24x^3$

Final Answer:

$h'(x) = 27x^{-4} + 24x^3$

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you differentiate terms with negative exponents?
2. What happens when differentiating a constant?
3. How do you simplify a function before differentiating?
4. How would the derivative change if the power was a fraction less than 1?
5. Can you differentiate a function that includes radicals in a different way?

Tip: Always simplify the expression before taking the derivative, as it can make the process easier.