Question

If \( x = f(t) \) and \( y = g(t) \) are differentiable functions of \( t \), so that \( y \) is a function of \( x \) and \( \frac{dx}{dt} \neq 0 \), then prove that: \[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}. \] Hence find \( \frac{dy}{dx} \), if \( x = at^2 \), \( y = 2at \).

Hint:

For parametric differentiation: \[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}. \]

Answer 1

Step 1: Proof of Chain Rule
Since \( y = g(t) \) and \( x = f(t) \), we differentiate both functions with respect to \( t \): \[ \frac{dx}{dt} = f'(t), \quad \frac{dy}{dt} = g'(t). \] Using the chain rule for derivatives: \[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}. \] Step 2: Compute \( \frac{dy}{dx} \) for Given Functions
Given: \[ x = at^2, \quad y = 2at. \] Differentiate with respect to \( t \): \[ \frac{dx}{dt} = 2at, \quad \frac{dy}{dt} = 2a. \] Using the chain rule: \[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{2a}{2at} = \frac{1}{t}. \]