Find the derivatives of the following (1-3) functions :
`y=5/2x^(7)`

Answer

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

### Step-by-Step Solution:

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

2. **Apply the constant multiple rule**:
The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. Here, the constant is \( \frac{5}{2} \) and the function is \( x^7 \).

3. **Differentiate using the power rule**:
According to the power rule, the derivative of \( x^7 \) is:
\[ \frac{d}{dx}(x^7) = 7x^{7-1} = 7x^6 \]

4. **Combine the results**:
Now, multiply the derivative of \( x^7 \) by the constant \( \frac{5}{2} \):
\[ \frac{dy}{dx} = \frac{5}{2} \cdot 7x^6 \]

5. **Simplify the expression**:
Multiply \( \frac{5}{2} \) by \( 7 \):
\[ \frac{dy}{dx} = \frac{35}{2} x^6 \]

### Final Answer:
\[ \frac{dy}{dx} = \frac{35}{2} x^6 \]