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Question

Find dydx, if x and y are connected parametrically by the equations given in questions without eliminating the parameter.

x=sin3tcost 2t,y=cos3tcos 2t

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Solution

Given, x=sin3tcost 2t,y=cos3tcos 2t

Differentiating w.r.t. t, we get

dxdt=ddt(sin3tcos 2t)=cos 2t(3 sin2t cos t)sin3t(2sin 2txcos 2t)(cos 2t)2 (Using quotient rule,ddx(uv)=vdudxududxv2)=3(cos 2t)sin2tcos t+sin 2tsin3tcos 2tcos 2t=3(12sin2t)sin2t cos t+(2sin t cos t)sin3tcos 2tcos 2t ( cos 2t=12sin2t and sin 2t=2sin t cos t)=3sin2t cos t4sin4t cos tcos 2tcos 2tand dydt=ddt(cos3tcos 2t)=cos 2t(3cos2 tsin t)cos3t(2sin 2t2cos 2t)(cos 2t)2 (Using quotient rule)=3(cos 2t)cos2t sin t+sin 2t cos3tcos 2tcos 2t=3(2cos2t1)cos2t sin t+cos3t(2sin t cos t)cos 2tcos 2t [cos 2t=2cos2t1 sin 2t=2sin t cos t]=3cos2t sin t4cos4t sin tcos 2tcos 2t dydx=dydtdxdt=dydt×dtdx=3cos2t sin t4cos4t sin t3sin2t cos t4sin4t cos t=cos2tsin t(34cos2t)sin2t cos t(34sin2t)=cos t(34cos2t)sin t(34sin2t)=3 cos t4cos3t3 sin t4 sin3t=3cos 3tsin 3t=cot 3t [cos 3t=4cos3t3cos tsin 3t=3 sin t4 sin3t]


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