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  • Find the second order derivative of  with respect to <img alt="\operatorname{asin}^3 t \t

Find the second order derivative of acos^{3}t with respect to \operatorname{asin}^3 t \text { at } t=\frac{\pi}{4}

Option: 1

\frac{2 \sqrt{2}}{3 a}


Option: 2

\frac{4 \sqrt{2}}{2 a}


Option: 3

\frac{4 \sqrt{2}}{3 a}


Option: 4

\frac{3 \sqrt{2}}{4 a}


Answers (1)

best_answer

Let y=a \cos ^3 tand x=a \sin ^3 t \begin{aligned} & \frac{d y}{d t}=-3 a \cos ^2 t \sin t \text { and } \frac{d x}{d t}=3 a \sin ^2 t \cos t \\ & \frac{d y}{d x}=\frac{d y}{\frac{d t}{d t}}=-\frac{3 a \cos ^2 t \sin t}{3 a \sin ^2 t \cos t}=-\cot (t) \end{aligned}

Differentiate it again,

\begin{aligned} & \frac{d^2 y}{d x}=\operatorname{cosec}^2 x \cdot \frac{d t}{d x}=\frac{\operatorname{cosec}^2 x}{3 a \sin ^2 t \cdot \cos t} \\ & {\left[\frac{d^2 y}{d x}\right]_{t=\frac{\pi}{4}}=\frac{2}{3 a \times \frac{1}{2} \times \frac{1}{\sqrt{2}}}=\frac{4 \sqrt{2}}{3 a}} \end{aligned}

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