 Find the derivatives of the function x^3/2 sin ^3 x + (sin x )^3/2

To find the derivative of the function \( f(x) = \frac{x^{3/2}}{\sin^3(x)} + \left(\sin(x)\right)^{3/2} \), we will use the rules of differentiation including the product rule, chain rule, and quotient rule where necessary.

### Function:
\[
f(x) = \frac{x^{3/2}}{\sin^3(x)} + \left( \sin(x) \right)^{3/2}
\]

### Derivative of \( f(x) \):
1. **First term: \( \frac{x^{3/2}}{\sin^3(x)} \):**

We apply the quotient rule. The quotient rule is:
\[
\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{v(x)^2}
\]

Here, \( u(x) = x^{3/2} \) and \( v(x) = \sin^3(x) \).

- Derivative of \( u(x) = x^{3/2} \):
\[
u'(x) = \frac{3}{2} x^{1/2}
\]

- Derivative of \( v(x) = \sin^3(x) \) (using the chain rule):
\[
v'(x) = 3 \sin^2(x) \cdot \cos(x)
\]

Now, applying the quotient rule:
\[
\frac{d}{dx} \left( \frac{x^{3/2}}{\sin^3(x)} \right) = \frac{\sin^3(x) \cdot \frac{3}{2} x^{1/2} - x^{3/2} \cdot 3 \sin^2(x) \cos(x)}{\sin^6(x)}
\]

Simplify the numerator:
\[
= \frac{\frac{3}{2} x^{1/2} \sin^3(x) - 3 x^{3/2} \sin^2(x) \cos(x)}{\sin^6(x)}
\]

2. **Second term: \( \left( \sin(x) \right)^{3/2} \):**

We apply the chain rule. The chain rule is:
\[
\frac{d}{dx} \left( g(h(x)) \right) = g'(h(x)) \cdot h'(x)
\]

Here, \( g(x) = x^{3/2} \) and \( h(x) = \sin(x) \).

- Derivative of \( g(x) = x^{3/2} \):
\[
g'(x) = \frac{3}{2} x^{1/2}
\]

- Derivative of \( h(x) = \sin(x) \):
\[
h'(x) = \cos(x)
\]

Now, applying the chain rule:
\[
\frac{d}{dx} \left( \sin(x) \right)^{3/2} = \frac{3}{2} \left( \sin(x) \right)^{1/2} \cdot \cos(x)
\]

### Final derivative:
Putting both terms together, we get:
\[
f'(x) = \frac{\frac{3}{2} x^{1/2} \sin^3(x) - 3 x^{3/2} \sin^2(x) \cos(x)}{\sin^6(x)} + \frac{3}{2} \left( \sin(x) \right)^{1/2} \cdot \cos(x)
\]

---

Would you like further details on simplifying the derivative or specific steps in the calculation? Here are some follow-up questions to expand your understanding:

1. How does the quotient rule work when applied to different types of functions?
2. What is the chain rule, and why is it useful in calculus?
3. Can you simplify the result further by factoring?
4. How does the cosine function affect the derivative in trigonometric expressions?
5. What happens to the derivative when evaluated at special angles, like \(x = 0\) or \(x = \pi\)?

**Tip:** Always double-check if functions are suitable for applying the chain, product, or quotient rule by breaking them down into simpler parts!

Find the derivatives of the function x^3/2 sin^3 x + (sin x)^3/2

Find the derivative of the function x^3/2 divided by sin^3(x) plus (sin(x))^3/2 using the quotient rule and chain rule. This detailed step-by-step solution explains how to handle complex expressions involving trigonometric and power functions, suitable for students in Grades 11-12 or early college calculus.

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