If ` x=2at ^(2),y =4at ,then (d^(2)y)/(dx^(2) )=`

Options

A. ` (-1)/( 2at^(3) )`
B. ` (1)/( 2at^(3) )`
C. ` (-1)/( 4at^(3) )`
D. ` (1)/( 4at^(3) )`

Answer

To solve the problem, we need to find the second derivative of \( y \) with respect to \( x \), given the equations \( x = 2at^2 \) and \( y = 4at \).

### Step-by-Step Solution:

1. **Find \( \frac{dy}{dt} \) and \( \frac{dx}{dt} \)**:
- Given \( y = 4at \), differentiate with respect to \( t \):
\[ \frac{dy}{dt} = 4a \]
- Given \( x = 2at^2 \), differentiate with respect to \( t \):
\[ \frac{dx}{dt} = 4at \]

2. **Find \( \frac{dy}{dx} \)**:
- Using the chain rule, we can express \( \frac{dy}{dx} \) as:
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{4a}{4at} = \frac{1}{t} \]

3. **Find \( \frac{d^2y}{dx^2} \)**:
- To find the second derivative, we need to differentiate \( \frac{dy}{dx} \) with respect to \( x \):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{d}{dx}\left(\frac{1}{t}\right) \]
- Using the chain rule:
\[ \frac{d}{dx}\left(\frac{1}{t}\right) = \frac{d}{dt}\left(\frac{1}{t}\right) \cdot \frac{dt}{dx} \]
- The derivative of \( \frac{1}{t} \) with respect to \( t \) is:
\[ \frac{d}{dt}\left(\frac{1}{t}\right) = -\frac{1}{t^2} \]

4. **Find \( \frac{dt}{dx} \)**:
- From \( \frac{dx}{dt} = 4at \), we have:
\[ \frac{dt}{dx} = \frac{1}{\frac{dx}{dt}} = \frac{1}{4at} \]

5. **Combine the results**:
- Now substituting back:
\[ \frac{d^2y}{dx^2} = -\frac{1}{t^2} \cdot \frac{1}{4at} = -\frac{1}{4at^3} \]

### Final Result:
\[ \frac{d^2y}{dx^2} = -\frac{1}{4at^3} \]