Find the derivatives of the following functions (1-3) at any point of their domains :
`f(x)=3x^(6)-x^(4)/4`

Answer

To find the derivative of the function \( f(x) = 3x^{6} - \frac{x^{4}}{4} \), we will use the power rule of differentiation. The power rule states that if \( f(x) = x^n \), then \( f'(x) = n \cdot x^{n-1} \).

### Step-by-Step Solution:

1. **Identify the function**:
\[ f(x) = 3x^{6} - \frac{x^{4}}{4} \]

2. **Differentiate the first term** \( 3x^{6} \):
- Using the power rule:
\[ \frac{d}{dx}(3x^{6}) = 3 \cdot 6x^{6-1} = 18x^{5} \]

3. **Differentiate the second term** \( -\frac{x^{4}}{4} \):
- First, rewrite it as \( -\frac{1}{4}x^{4} \).
- Now apply the power rule:
\[ \frac{d}{dx}\left(-\frac{1}{4}x^{4}\right) = -\frac{1}{4} \cdot 4x^{4-1} = -x^{3} \]

4. **Combine the derivatives**:
\[ f'(x) = 18x^{5} - x^{3} \]

5. **Final result**:
\[ f'(x) = 18x^{5} - x^{3} \]