If `x=2at^(2),y=at^(4),"then "dy/dx=t^(2)`.

Answer

To solve the problem, we need to find \(\frac{dy}{dx}\) given the parametric equations \(x = 2at^2\) and \(y = at^4\).

### Step-by-step Solution:

1. **Differentiate \(x\) with respect to \(t\)**:
\[ x = 2at^2 \]
Differentiate both sides with respect to \(t\):
\[ \frac{dx}{dt} = \frac{d}{dt}(2at^2) = 2a \cdot 2t = 4at \]

2. **Differentiate \(y\) with respect to \(t\)**:
\[ y = at^4 \]
Differentiate both sides with respect to \(t\):
\[ \frac{dy}{dt} = \frac{d}{dt}(at^4) = a \cdot 4t^3 = 4at^3 \]

3. **Find \(\frac{dy}{dx}\)** using the chain rule:
The chain rule states that:
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \]
Substitute the derivatives we found:
\[ \frac{dy}{dx} = \frac{4at^3}{4at} = \frac{t^3}{t} = t^2 \]

4. **Conclusion**:
Therefore, we have shown that:
\[ \frac{dy}{dx} = t^2 \]