Find the derivatives of the following functions (1-3) at any point of their domains :
`y=1/7x^(7)+1/5x^(5)-2/3x^(3)+5`

Answer

To find the derivative of the function \( y = \frac{1}{7}x^7 + \frac{1}{5}x^5 - \frac{2}{3}x^3 + 5 \), we will use the power rule of differentiation. The power rule states that if \( y = x^n \), then \( \frac{dy}{dx} = nx^{n-1} \).

### Step-by-Step Solution:

1. **Identify the function**:
\[ y = \frac{1}{7}x^7 + \frac{1}{5}x^5 - \frac{2}{3}x^3 + 5 \]

2. **Differentiate each term**:
- For the first term \( \frac{1}{7}x^7 \):
\[ \frac{d}{dx}\left(\frac{1}{7}x^7\right) = \frac{1}{7} \cdot 7x^{7-1} = x^6 \]
- For the second term \( \frac{1}{5}x^5 \):
\[ \frac{d}{dx}\left(\frac{1}{5}x^5\right) = \frac{1}{5} \cdot 5x^{5-1} = x^4 \]
- For the third term \( -\frac{2}{3}x^3 \):
\[ \frac{d}{dx}\left(-\frac{2}{3}x^3\right) = -\frac{2}{3} \cdot 3x^{3-1} = -2x^2 \]
- For the constant term \( 5 \):
\[ \frac{d}{dx}(5) = 0 \]

3. **Combine the derivatives**:
\[ \frac{dy}{dx} = x^6 + x^4 - 2x^2 + 0 \]
Simplifying this gives:
\[ \frac{dy}{dx} = x^6 + x^4 - 2x^2 \]

4. **Factor the derivative if possible**:
We can factor out \( x^2 \):
\[ \frac{dy}{dx} = x^2(x^4 + x^2 - 2) \]

### Final Answer:
The derivative of the function \( y \) is:
\[ \frac{dy}{dx} = x^2(x^4 + x^2 - 2) \]