Find $\frac{d y}{d x}$, if $x=a t^{2}, y=2 a t$.

Step 1: Differentiate x with respect to t

We are given $x$ as a function of $t$, specifically $x = at^2$. To find $\frac{dx}{dt}$, we differentiate $at^2$ with respect to $t$. Using the power rule for differentiation, $\frac{d}{dt}(t^n) = nt^{n-1}$, we get $2at$.

Step 2: Differentiate y with respect to t

Next, we differentiate $y$ with respect to $t$. Given $y = 2at$, its derivative with respect to $t$ is $2a$, as $a$ is a constant and the derivative of $t$ with respect to $t$ is $1$.

Step 3: Apply the Chain Rule

To find $\frac{dy}{dx}$, we use the chain rule for parametric equations. This rule states that $\frac{dy}{dx}$ can be found by dividing $\frac{dy}{dt}$ by $\frac{dx}{dt}$.

Step 4: Substitute and Simplify

Now, we substitute the expressions we found for $\frac{dy}{dt}$ and $\frac{dx}{dt}$ into the chain rule formula. After substitution, we simplify the expression by canceling out common terms.

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