Find the derivatives of the following (9-10) at any point of their domains :
`f(x)=x^(5)(3-6x^(-9))`

Answer

To find the derivative of the function \( f(x) = x^5(3 - 6x^{-9}) \), we will use the product rule and the power rule of differentiation.

### Step-by-Step Solution:

1. **Identify the function**:
\[ f(x) = x^5(3 - 6x^{-9}) \]

2. **Apply the product rule**:
The product rule states that if you have two functions \( u(x) \) and \( v(x) \), then the derivative \( (uv)' = u'v + uv' \). Here, let:
- \( u = x^5 \)
- \( v = 3 - 6x^{-9} \)

3. **Differentiate \( u \) and \( v \)**:
- The derivative of \( u \):
\[ u' = \frac{d}{dx}(x^5) = 5x^4 \]
- The derivative of \( v \):
\[ v' = \frac{d}{dx}(3 - 6x^{-9}) = 0 - 6 \cdot (-9)x^{-10} = 54x^{-10} \]

4. **Apply the product rule**:
Now, substitute \( u, u', v, \) and \( v' \) into the product rule:
\[ f'(x) = u'v + uv' = (5x^4)(3 - 6x^{-9}) + (x^5)(54x^{-10}) \]

5. **Simplify the expression**:
- First term:
\[ 5x^4(3 - 6x^{-9}) = 15x^4 - 30x^{-5} \]
- Second term:
\[ x^5(54x^{-10}) = 54x^{-5} \]

Combine both terms:
\[ f'(x) = 15x^4 - 30x^{-5} + 54x^{-5} \]
\[ f'(x) = 15x^4 + (54 - 30)x^{-5} = 15x^4 + 24x^{-5} \]

6. **Final result**:
Thus, the derivative of the function \( f(x) \) is:
\[ f'(x) = 15x^4 + 24x^{-5} \]