If `x = t^(2)` and `y = t^(3)`, then `(d^(2)y)/(dx^(2))` is equal to

Options

A. `3/2`
B. `3/(4t)`
C. `3/(2t)`
D. `(3)/(2t)`

Explanation

To find \(\frac{d^2y}{dx^2}\) given \(x = t^2\) and \(y = t^3\), we will follow these steps:

### Step 1: Differentiate \(x\) and \(y\) with respect to \(t\)
1. Start with the equations:
\[ x = t^2 \quad \text{and} \quad y = t^3 \]
2. Differentiate \(x\) with respect to \(t\):
\[ \frac{dx}{dt} = 2t \]
3. Differentiate \(y\) with respect to \(t\):
\[ \frac{dy}{dt} = 3t^2 \]

### Step 2: Find \(\frac{dy}{dx}\)
4. Use the chain rule to find \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{3t^2}{2t} = \frac{3t}{2} \]

### Step 3: Differentiate \(\frac{dy}{dx}\) with respect to \(x\)
5. To find \(\frac{d^2y}{dx^2}\), we need to differentiate \(\frac{dy}{dx}\) with respect to \(x\):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) \]
6. Since \(\frac{dy}{dx} = \frac{3t}{2}\), we can express this differentiation in terms of \(t\):
\[ \frac{d^2y}{dx^2} = \frac{d}{dt}\left(\frac{3t}{2}\right) \cdot \frac{dt}{dx} \]

### Step 4: Calculate \(\frac{dt}{dx}\)
7. From our earlier calculation, we know:
\[ \frac{dx}{dt} = 2t \implies \frac{dt}{dx} = \frac{1}{2t} \]

### Step 5: Substitute and simplify
8. Now substitute \(\frac{dt}{dx}\) into the expression for \(\frac{d^2y}{dx^2}\):
\[ \frac{d^2y}{dx^2} = \left(\frac{3}{2}\right) \cdot \frac{1}{2t} = \frac{3}{4t} \]

### Final Answer
Thus, the final result is:
\[ \frac{d^2y}{dx^2} = \frac{3}{4t} \]