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  • Find the second order derivative if x and y are given by  and <img alt="y=acost" src="https://ent

Find the second order derivative if x and y are given by x=asint and y=acost

Option: 1

\frac{1}{a} \cos ^3 t


Option: 2

-\frac{1}{a} \cos ^3 t


Option: 3

\frac{1}{a} \sec ^3 t


Option: 4

-\frac{1}{a} \sec ^3 t


Answers (1)

best_answer

Differentiating the function implicitly with respect to “x”;


\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\frac{d y}{d t}}{\frac{\mathrm{d} x}{d t}}=\frac{-a \sin t}{a \cos t}=-\frac{\sin t}{\cos t}

Again differentiating with respect to “x”;


\begin{aligned} & \frac{d^2 y}{d x^2}=\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{d y}{d x}\right)=\frac{\mathrm{d}}{\mathrm{d} t}\left(-\frac{\sin t}{\cos t}\right) \frac{d t}{d x} \\ & \frac{d^2 y}{d x^2}=-\left(\sec ^2 t\right) \times \frac{1}{a \cos t}=-\frac{\sec ^2 t}{a \cos t}=-\frac{1}{a} \sec ^3 t \end{aligned}

Posted by

Rakesh

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