Method of Differentiation
Parametric Differentiation
Premium Question
Grade 12
Question:
If $x = a \cos^3 t$ and $y = a \sin^3 t$, then the value of $\frac{d^2y}{dx^2}$ at $t = \frac{\pi}{4}$ is:
(1) $\frac{4\sqrt{2}}{3a}$
(2) $\frac{2\sqrt{2}}{3a}$
(3) $-\frac{4\sqrt{2}}{3a}$
(4) $\frac{4}{3a}$
Step-by-Step Solution
Key Concept: Show that $\frac{dy}{dx}$ simplifies neatly to $-\tan t$, then differentiate this with respect to $t$ and divide once more by $\frac{dx}{dt}$ before substituting $t = \pi/4$.
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Correct Answer: (1)