Limits, Continuity & Differentiability
Parametric Functions
Grade 12
Question:
<p>If \(x = f(t), y = y(t)\), then \(\frac{d^2y}{dx^2} =\)</p>
<p>(a) \(\frac{f'y'' - y'f''}{(f')^2}\)</p>
<p>(b) \(\frac{f'y'' - y'f''}{(f')^3}\)</p>
<p>(c) \(\frac{y''}{f'} - \frac{y'f''}{(f')^2}\)</p>
<p>(d) \(\frac{y''}{(f')^2} - \frac{y'f''}{(f')^3}\)</p>
Step-by-Step Solution
Key Concept: Apply quotient rule for differentiation of parametric functions.
<p>For parametric functions, $\frac{dy}{dx} = \frac{y'}{f'}$.</p><p>Then $\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{y'}{f'}\right) = \frac{d/dt\left(\frac{y'}{f'}\right)}{dx/dt} = \frac{f'y'' - y'f''}{(f')^3}$</p>
Correct Answer: B