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

Answer

To find \(\frac{d^2y}{dx^2}\) given the equations \(x = at^2\) and \(y = 2at\), we will follow these steps:

### Step 1: Differentiate \(y\) with respect to \(t\)
Given:
\[ y = 2at \]
Differentiating \(y\) with respect to \(t\):
\[ \frac{dy}{dt} = 2a \]

### Step 2: Differentiate \(x\) with respect to \(t\)
Given:
\[ x = at^2 \]
Differentiating \(x\) with respect to \(t\):
\[ \frac{dx}{dt} = 2at \]

### Step 3: Find \(\frac{dy}{dx}\)
Using the chain rule, we can express \(\frac{dy}{dx}\) as:
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \]
Substituting the values from Steps 1 and 2:
\[ \frac{dy}{dx} = \frac{2a}{2at} = \frac{1}{t} \]

### Step 4: Differentiate \(\frac{dy}{dx}\) with respect to \(x\) to find \(\frac{d^2y}{dx^2}\)
We need to differentiate \(\frac{dy}{dx} = \frac{1}{t}\) with respect to \(x\). First, we need to express \(\frac{dt}{dx}\):
From \(\frac{dx}{dt} = 2at\), we can find \(\frac{dt}{dx}\):
\[ \frac{dt}{dx} = \frac{1}{2at} \]
Now, using the chain rule:
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{d}{dt}\left(\frac{dy}{dx}\right) \cdot \frac{dt}{dx} \]
Differentiating \(\frac{dy}{dx} = \frac{1}{t}\) with respect to \(t\):
\[ \frac{d}{dt}\left(\frac{1}{t}\right) = -\frac{1}{t^2} \]
Now substituting this back:
\[ \frac{d^2y}{dx^2} = -\frac{1}{t^2} \cdot \frac{1}{2at} = -\frac{1}{2at^3} \]

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