Limits, Continuity & Differentiability
Methods of Differentiation
Grade 12
Question:
<p>If $x = t^2 + t + 5$ and $y = \sin t$, then $\dfrac{d^2y}{dx^2}$ is:</p>
<p>$-\dfrac{(3t^2+1)\sin t + 6\cos t}{(3t^2+1)^3}$</p>
<p>$\dfrac{(3t^2+1)\sin t + 6t\cos t}{(3t^2+1)^3}$</p>
<p>$-\dfrac{(3t^2+1)\sin t + 6t\cos t}{(3t^2+1)^3}$</p>
<p>$\dfrac{\cos t}{3t^2+1}$</p>
Step-by-Step Solution
Key Concept: General
<b>Parametric Second Derivative</b><br>$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{\cos t}{2t+1}$<br>$\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{\frac{d}{dt}\left(\frac{\cos t}{2t+1}\right)}{\frac{dx}{dt}}$<br>$\frac{d}{dt}\left(\frac{\cos t}{2t+1}\right) = \frac{-\sin t(2t+1) - 2\cos t}{(2t+1)^2}$<br>$\frac{d^2y}{dx^2} = \frac{-\sin t(2t+1) - 2\cos t}{(2t+1)^3}$<br><b>Key concept:</b> $\frac{d^2y}{dx^2} = \frac{(d/dt)(dy/dx)}{dx/dt}$, NOT $(d^2y/dt^2)/(d^2x/dt^2)$.<br><b>Trap:</b> Dividing second derivatives directly is the most common error in parametric differentiation.
Correct Answer: A