If $x = t^2$ and $y = t^3$, then $\frac{d^2 y}{dx^2}$ is equal to :

Step 1: Find the first derivative $\frac{dy}{dx}$

We are given $x$ and $y$ in terms of a parameter $t$. To find $\frac{dy}{dx}$, we use the chain rule for parametric equations. First, we find the derivatives of $x$ and $y$ with respect to $t$.

Step 2: Calculate $\frac{dx}{dt}$ and $\frac{dy}{dt}$

Differentiating $x = t^2$ with respect to $t$ gives $2t$. Differentiating $y = t^3$ with respect to $t$ gives $3t^2$.

Step 3: Substitute to find $\frac{dy}{dx}$

Now we substitute the derivatives $\frac{dx}{dt}$ and $\frac{dy}{dt}$ into the chain rule formula to find $\frac{dy}{dx}$. We simplify the expression by canceling out $t$.

Step 4: Find the second derivative $\frac{d^2 y}{dx^2}$

To find the second derivative $\frac{d^2 y}{dx^2}$, we need to differentiate $\frac{dy}{dx}$ with respect to $x$. Since $\frac{dy}{dx}$ is a function of $t$, we again use the chain rule for parametric differentiation.

Step 5: Calculate $\frac{d}{dt}\left(\frac{dy}{dx}\right)$ and substitute

First, we differentiate $\frac{dy}{dx} = \frac{3}{2}t$ with respect to $t$, which gives $\frac{3}{2}$. Then, we divide this result by $\frac{dx}{dt} = 2t$ to obtain the second derivative $\frac{d^2 y}{dx^2}$.

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