If x= 5m, y= m , where m is parameter, then `(dy)/(dx) = 1/5`. (True/False)

Answer

To solve the problem, we need to determine if \( \frac{dy}{dx} = \frac{1}{5} \) given \( x = 5m \) and \( y = m \), where \( m \) is a parameter.

### Step-by-Step Solution:

1. **Identify the equations:**
\[ x = 5m \quad \text{and} \quad y = m \]

2. **Differentiate \( x \) with respect to \( m \):**
\[ \frac{dx}{dm} = \frac{d(5m)}{dm} = 5 \cdot \frac{dm}{dm} = 5 \]

3. **Differentiate \( y \) with respect to \( m \):**
\[ \frac{dy}{dm} = \frac{d(m)}{dm} = 1 \]

4. **Use the chain rule to find \( \frac{dy}{dx} \):**
\[ \frac{dy}{dx} = \frac{dy}{dm} \cdot \frac{dm}{dx} \]

5. **Find \( \frac{dm}{dx} \):**
Since \( \frac{dx}{dm} = 5 \), we can find \( \frac{dm}{dx} \) as the reciprocal:
\[ \frac{dm}{dx} = \frac{1}{\frac{dx}{dm}} = \frac{1}{5} \]

6. **Substituting the values into the chain rule:**
\[ \frac{dy}{dx} = \frac{dy}{dm} \cdot \frac{dm}{dx} = 1 \cdot \frac{1}{5} = \frac{1}{5} \]

7. **Conclusion:**
Since we have found that \( \frac{dy}{dx} = \frac{1}{5} \), we can confirm that the statement is **True**.