Find the derivatives of the following functions (1-3) at any point of their domains :
`G(t)=-2t^(10)+1-t^(3)+7t^(5)`

Answer

To find the derivative of the function \( G(t) = -2t^{10} + 1 - t^{3} + 7t^{5} \), we will apply the power rule for differentiation. The power rule states that if \( f(t) = t^n \), then the derivative \( f'(t) = n \cdot t^{n-1} \).

### Step-by-Step Solution:

1. **Identify the function**:
\[ G(t) = -2t^{10} + 1 - t^{3} + 7t^{5} \]

2. **Differentiate each term**:
- The derivative of \(-2t^{10}\):
\[ \frac{d}{dt}(-2t^{10}) = -2 \cdot 10t^{10-1} = -20t^{9} \]
- The derivative of the constant \(1\):
\[ \frac{d}{dt}(1) = 0 \]
- The derivative of \(-t^{3}\):
\[ \frac{d}{dt}(-t^{3}) = -3t^{3-1} = -3t^{2} \]
- The derivative of \(7t^{5}\):
\[ \frac{d}{dt}(7t^{5}) = 7 \cdot 5t^{5-1} = 35t^{4} \]

3. **Combine the derivatives**:
Now, we combine all the derivatives we calculated:
\[ G'(t) = -20t^{9} + 0 - 3t^{2} + 35t^{4} \]

4. **Simplify the expression**:
\[ G'(t) = -20t^{9} + 35t^{4} - 3t^{2} \]

### Final Answer:
Thus, the derivative of the function \( G(t) \) is:
\[ G'(t) = -20t^{9} + 35t^{4} - 3t^{2} \]