Question

When \( x = at^2 \), \( y = 2at \), then \( \frac{dy}{dx} \) is:

• A. \( \frac{1}{t} \)
• B. \( \frac{3}{t} \)
• C. \( \frac{2}{t} \)
• D. None of these

Hint:

When differentiating functions involving multiple variables, use the chain rule to find \( \frac{dy}{dx} \) by differentiating with respect to the intermediate variable and then applying the relationship between the variables.

Answer 1

The Correct Option is C

Step 1: Differentiating using the chain rule. 
We are given \( x = at^2 \) and \( y = 2at \). To find \( \frac{dy}{dx} \), we use the chain rule. First, we find \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \). 
Step 2: Calculating \( \frac{dx}{dt} \). 
Differentiating \( x = at^2 \) with respect to \( t \): \[ \frac{dx}{dt} = 2at \] Step 3: Calculating \( \frac{dy}{dt} \). 
Differentiating \( y = 2at \) with respect to \( t \): \[ \frac{dy}{dt} = 2a \] Step 4: Applying the chain rule. 
Now, using the chain rule: \[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{2a}{2at} = \frac{1}{t} \] Step 5: Conclusion. 
Thus, \( \frac{dy}{dx} = \frac{2}{t} \), corresponding to option (c).