Differentiate the following w.r.t. x:
`(3x^(2)-9x+5)^(9)`

Answer

To differentiate the function \( y = (3x^2 - 9x + 5)^9 \) with respect to \( x \), we will use the chain rule. The chain rule states that if you have a composite function \( y = f(g(x)) \), then the derivative \( \frac{dy}{dx} \) is given by:

\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]

### Step-by-Step Solution:

1. **Identify the outer and inner functions**:
- Let \( u = 3x^2 - 9x + 5 \) (inner function).
- Then, \( y = u^9 \) (outer function).

2. **Differentiate the outer function**:
- The derivative of \( y = u^9 \) with respect to \( u \) is:
\[ \frac{dy}{du} = 9u^8 \]

3. **Differentiate the inner function**:
- Now, differentiate \( u = 3x^2 - 9x + 5 \) with respect to \( x \):
\[ \frac{du}{dx} = 6x - 9 \]

4. **Apply the chain rule**:
- Now, using the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 9u^8 \cdot (6x - 9) \]

5. **Substitute back the value of \( u \)**:
- Substitute \( u = 3x^2 - 9x + 5 \) back into the equation:
\[ \frac{dy}{dx} = 9(3x^2 - 9x + 5)^8 \cdot (6x - 9) \]

6. **Simplify the expression**:
- You can factor out the constants:
\[ \frac{dy}{dx} = 9(6x - 9)(3x^2 - 9x + 5)^8 \]

### Final Answer:
\[ \frac{dy}{dx} = 9(6x - 9)(3x^2 - 9x + 5)^8 \]