Find the derivative for the given function. Write your answer using positive and negative exponents and fractional exponents instead of radicals.

h(x)=(5x−3−5x+8)13−9x−3+6x+4

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.

Find the derivative for the given function. Write your answer using positive and negative exponents and fractional exponents instead of radicals. h(x) = (5x - 3 - 5x + 8)^(1/3) - 9x^(-3) + 6x^4

Learn how to differentiate the function h(x) = (5x - 3 - 5x + 8)^(1/3) - 9x^(-3) + 6x^4 using positive, negative, and fractional exponents. This solution demonstrates the power rule for differentiation and simplification techniques, suitable for Grades 10-12 students.

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