If ` x= at^(4) `
`y = 2at^(2)` then `(dy)/(dx)=` _____

Options

A. ` 1//t`
B. ` -1//t`
C. ` 1//t^(2)`
D. ` -1//t^(2)`

Answer

To find \(\frac{dy}{dx}\) given the parametric equations \(x = at^4\) and \(y = 2at^2\), we will use the chain rule.

### Step-by-Step Solution:

1. **Differentiate \(x\) with respect to \(t\)**:
\[ x = at^4 \]
To find \(\frac{dx}{dt}\), we differentiate \(x\) with respect to \(t\):
\[ \frac{dx}{dt} = \frac{d}{dt}(at^4) = 4at^3 \]

2. **Differentiate \(y\) with respect to \(t\)**:
\[ y = 2at^2 \]
To find \(\frac{dy}{dt}\), we differentiate \(y\) with respect to \(t\):
\[ \frac{dy}{dt} = \frac{d}{dt}(2at^2) = 4at \]

3. **Apply the chain rule to find \(\frac{dy}{dx}\)**:
The chain rule states that:
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = \frac{4at}{4at^3} \]

4. **Simplify the expression**:
\[ \frac{dy}{dx} = \frac{4at}{4at^3} = \frac{1}{t^2} \]

Thus, the final answer is:
\[ \frac{dy}{dx} = \frac{1}{t^2} \]