If `x=t^(2)andy=t^(3)` , then `(d^(2)y)/(dx^(2))` is equal to: a) `(3)/(2)` b) `(3)/(2)t` c) `(3)/(2t)` d) `(3)/(4t)`

Options

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

Answer

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

### Step 1: Find \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\)

Given:
- \(x = t^2\)
- \(y = t^3\)

We differentiate both \(x\) and \(y\) with respect to \(t\):

\[ \frac{dx}{dt} = \frac{d(t^2)}{dt} = 2t \]

\[ \frac{dy}{dt} = \frac{d(t^3)}{dt} = 3t^2 \]

**Hint for Step 1:** Differentiate \(x\) and \(y\) with respect to \(t\) using the power rule.

### Step 2: Find \(\frac{dy}{dx}\)

Using the chain rule, we can express \(\frac{dy}{dx}\) in terms of \(\frac{dy}{dt}\) and \(\frac{dx}{dt}\):

\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{3t^2}{2t} \]

Simplifying this gives:

\[ \frac{dy}{dx} = \frac{3}{2}t \]

**Hint for Step 2:** Use the relationship \(\frac{dy}{dx} = \frac{dy/dt}{dx/dt}\) to find the first derivative.

### Step 3: Find \(\frac{d^2y}{dx^2}\)

To find the second derivative \(\frac{d^2y}{dx^2}\), we need to differentiate \(\frac{dy}{dx}\) with respect to \(x\):

Using the chain rule again, we have:

\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{d}{dt}\left(\frac{dy}{dx}\right) \cdot \frac{dt}{dx} \]

We already have \(\frac{dy}{dx} = \frac{3}{2}t\). Now we differentiate this with respect to \(t\):

\[ \frac{d}{dt}\left(\frac{3}{2}t\right) = \frac{3}{2} \]

Next, we need \(\frac{dt}{dx}\). We know:

\[ \frac{dx}{dt} = 2t \implies \frac{dt}{dx} = \frac{1}{2t} \]

Now substituting back into the second derivative formula:

\[ \frac{d^2y}{dx^2} = \frac{3}{2} \cdot \frac{1}{2t} = \frac{3}{4t} \]

### Final Answer

Thus, we find that:

\[ \frac{d^2y}{dx^2} = \frac{3}{4t} \]

The correct option is **d) \(\frac{3}{4t}\)**.

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