Find `dy/dx` if :
`y=at^(2),t=x/(2a)`

Answer

To find \(\frac{dy}{dx}\) given \(y = at^2\) and \(t = \frac{x}{2a}\), we can follow these steps:

### Step 1: Substitute \(t\) in terms of \(x\) into the equation for \(y\)
We have:
\[ t = \frac{x}{2a} \]
Substituting this into the equation for \(y\):
\[ y = a\left(\frac{x}{2a}\right)^2 \]

### Step 2: Simplify the expression for \(y\)
Calculating \(y\):
\[ y = a\left(\frac{x^2}{(2a)^2}\right) = a\left(\frac{x^2}{4a^2}\right) = \frac{ax^2}{4a^2} \]
Now, simplifying further:
\[ y = \frac{x^2}{4a} \]

### Step 3: Differentiate \(y\) with respect to \(x\)
Now we differentiate \(y\) with respect to \(x\):
\[ \frac{dy}{dx} = \frac{d}{dx}\left(\frac{x^2}{4a}\right) \]
Using the power rule:
\[ \frac{dy}{dx} = \frac{1}{4a} \cdot 2x = \frac{2x}{4a} = \frac{x}{2a} \]

### Final Answer
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{x}{2a} \]

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