`(x^(4))/(sinx)`

Answer

To differentiate the function \( f(x) = \frac{x^4}{\sin x} \) with respect to \( x \), we will use the quotient rule. The quotient rule states that if you have a function in the form \( \frac{u}{v} \), the derivative is given by:

\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \]

### Step-by-Step Solution:

1. **Identify \( u \) and \( v \)**:
- Let \( u = x^4 \)
- Let \( v = \sin x \)

2. **Differentiate \( u \) and \( v \)**:
- The derivative of \( u \) is:
\[ \frac{du}{dx} = 4x^3 \]
- The derivative of \( v \) is:
\[ \frac{dv}{dx} = \cos x \]

3. **Apply the Quotient Rule**:
- Substitute \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \) into the quotient rule formula:
\[ \frac{d}{dx}\left(\frac{x^4}{\sin x}\right) = \frac{\sin x \cdot 4x^3 - x^4 \cdot \cos x}{(\sin x)^2} \]

4. **Simplify the Expression**:
- The expression can be rewritten as:
\[ \frac{4x^3 \sin x - x^4 \cos x}{\sin^2 x} \]

5. **Factor Out Common Terms**:
- Notice that \( 4x^3 \) can be factored out:
\[ = \frac{x^3 (4 \sin x - x \cos x)}{\sin^2 x} \]

### Final Answer:
The derivative of \( f(x) = \frac{x^4}{\sin x} \) is:
\[ f'(x) = \frac{x^3 (4 \sin x - x \cos x)}{\sin^2 x} \]