The curve \(C\) has equation \(x^2\tan y=9\), where \(0<y<\dfrac\pi2\) and \(x\neq0\).
Find \(\dfrac{\mathrm dy}{\mathrm dx}\), simplifying your answer in terms of \(x\).[4]
Given that \(\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{54(x^4-27)}{(x^4+81)^2}\), explain why the curve has a point of inflexion at \(x=\sqrt[4]{27}\).[3]
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Answer:(a) \(-\dfrac{18x}{x^4+81}\) (b) The second derivative changes from negative to positive at \(x=\sqrt[4]{27}\).