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To find the derivative \(\frac{dh}{ds}\) for the function 

\[
h = \sqrt[3]{\left(5s^2 - 9s\right)^6} + \sqrt{\left(5s^2 - 9s\right)^7}
\]

we can proceed by differentiating each term separately. Let’s denote the terms as:

1. \( h_1 = \sqrt[3]{\left(5s^2 - 9s\right)^6} \)
2. \( h_2 = \sqrt{\left(5s^2 - 9s\right)^7} \)

**Step-by-Step Solution:**

### 1. Differentiate \( h_1 = \sqrt[3]{\left(5s^2 - 9s\right)^6} \)
We can rewrite \( h_1 \) as:

\[
h_1 = \left(5s^2 - 9s\right)^2
\]

Then, the derivative of \( h_1 \) is:

\[
\frac{dh_1}{ds} = 2\left(5s^2 - 9s\right) \cdot \frac{d}{ds}(5s^2 - 9s)
\]

\[
\frac{dh_1}{ds} = 2\left(5s^2 - 9s\right) \cdot (10s - 9)
\]

### 2. Differentiate \( h_2 = \sqrt{\left(5s^2 - 9s\right)^7} \)
We can rewrite \( h_2 \) as:

\[
h_2 = \left(5s^2 - 9s\right)^{7/2}
\]

Then, the derivative of \( h_2 \) is:

\[
\frac{dh_2}{ds} = \frac{7}{2}\left(5s^2 - 9s\right)^{5/2} \cdot \frac{d}{ds}(5s^2 - 9s)
\]

\[
\frac{dh_2}{ds} = \frac{7}{2}\left(5s^2 - 9s\right)^{5/2} \cdot (10s - 9)
\]

### 3. Combine the derivatives

Now, sum up the derivatives \(\frac{dh_1}{ds}\) and \(\frac{dh_2}{ds}\) to get \(\frac{dh}{ds}\).

\[
\frac{dh}{ds} = 2\left(5s^2 - 9s\right)(10s - 9) + \frac{7}{2}\left(5s^2 - 9s\right)^{5/2} (10s - 9)
\]

This is the derivative of \( h \) with respect to \( s \).

Would you like a detailed expansion of the solution or any specific part of this derivative explained further?

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Here are 5 related questions to expand on this topic:

1. How do you find the derivative of a function raised to a power?
2. What is the chain rule and how is it applied in differentiation?
3. Can you find the second derivative of \( h \) with respect to \( s \)?
4. How do you differentiate functions that involve square roots or cube roots?
5. What are the rules for differentiating products and quotients of functions?

**Tip:** When differentiating complex expressions, always simplify the terms as much as possible before applying the differentiation rules.

Find the derivative dh/ds for h = cube root[(5s^2 - 9s)^6] + square root[(5s^2 - 9s)^7]

Learn how to find the derivative of a function involving a cube root and square root: h = cube root[(5s^2 - 9s)^6] + square root[(5s^2 - 9s)^7]. This problem uses the chain rule and power rule, and is suitable for students in advanced high school or undergraduate calculus.

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