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If x=acostheta,y=bsintheta, then prove t...

If `x=acostheta,y=bsintheta,` then prove that `(d^3y)/(dx^3)= -(3b)/(a^3)cos e c^4thetacotthetadot`

Text Solution

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We have `y=b sin theta, x = a cos theta.`
`"Therefore "(dy)/(dx)=((dy)/(d""theta))/((dx)/(d""theta))=-(b)/(a)cot theta`
`"or "(d^(2)y)/(dx^(2))=(b)/(a)cosec^(2)theta(d""theta)/(dx)=-(b)/(a^(2))cosec^(3)theta`
`"or "(d^(3)y)/(dx^(3))=-(b)/(a^(2))3cosec^(2)theta(-cosec theta cot theta) (d""theta)/(dx)`
`(3b)/(a^(2))cosec^(3)thetacot theta xx(-1)/(a sin theta)=-(3b)/(a^(3))cosec ^(4)theta cot theta`
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