Question

If $ x = f(\theta) $, $ y = g(\theta) $, then find: $$ \frac{d^2 y}{dx^2} $$

• A. \( \frac{g''(\theta)}{f'(\theta)} \)
• B. \( \frac{f''(\theta)}{x(\theta)} \)
• C. \( \frac{f'(\theta) g''(\theta) - g'(\theta) f''(\theta)}{(f'(\theta))^3} \)
• D. \( \frac{g'(\theta) f''(\theta) - g''(\theta) f'(\theta)}{(g'(\theta))^3} \)

Hint:

Use the chain rule for parametric derivatives: \[ \frac{d^2 y}{dx^2} = \frac{d}{d\theta} \left( \frac{dy}{dx} \right) \cdot \frac{1}{dx/d\theta} \]

Answer 1

The Correct Option is C

We are given parametric equations: \[ x = f(\theta),\ y = g(\theta) \] Then: \[ \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} = \frac{g'}{f'} \] Now, \[ \frac{d^2y}{dx^2} = \frac{d}{dx} \left( \frac{dy}{dx} \right) = \frac{d}{d\theta} \left( \frac{g'}{f'} \right) \cdot \frac{1}{dx/d\theta} \] Use quotient rule: \[ \frac{d}{d\theta} \left( \frac{g'}{f'} \right) = \frac{f' g'' - g' f''}{(f')^2} \] So: \[ \frac{d^2y}{dx^2} = \frac{f' g'' - g' f''}{(f')^2} \cdot \frac{1}{f'} = \boxed{ \frac{f' g'' - g' f''}{(f')^3} } \]