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

Options

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

Explanation

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

### Step 1: Differentiate \(y\) with respect to \(t\)
We start by differentiating \(y\) with respect to \(t\):
\[ \frac{dy}{dt} = \frac{d(at)}{dt} = a \]

### Step 2: Differentiate \(x\) with respect to \(t\)
Next, we differentiate \(x\) with respect to \(t\):
\[ \frac{dx}{dt} = \frac{d(at^2)}{dt} = 2at \]

### Step 3: Find \(\frac{dy}{dx}\)
Using the chain rule for parametric equations, we can find \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{a}{2at} = \frac{1}{2t} \]

### Step 4: Differentiate \(\frac{dy}{dx}\) with respect to \(t\)
Now we differentiate \(\frac{dy}{dx}\) with respect to \(t\):
\[ \frac{d}{dt}\left(\frac{dy}{dx}\right) = \frac{d}{dt}\left(\frac{1}{2t}\right) = -\frac{1}{2t^2} \]

### Step 5: Find \(\frac{dx}{dt}\) again
We already found \(\frac{dx}{dt} = 2at\).

### Step 6: Use the chain rule to find \(\frac{d^2y}{dx^2}\)
Now we can find \(\frac{d^2y}{dx^2}\) using the relation:
\[ \frac{d^2y}{dx^2} = \frac{d}{dt}\left(\frac{dy}{dx}\right) \cdot \frac{1}{\frac{dx}{dt}} \]
Substituting the values we have:
\[ \frac{d^2y}{dx^2} = \left(-\frac{1}{2t^2}\right) \cdot \frac{1}{2at} = -\frac{1}{4at^3} \]

### Final Answer
Thus, the final answer is:
\[ \frac{d^2y}{dx^2} = -\frac{1}{4at^3} \]

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