What is the derivative of #f(t) = (t/sint , cost/t^2 ) #?

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Answer 1 · 2017-07-15T04:57:56

#(dy)/(dx)=(sin^2t(tsint+2cost))/(t^3(sint-tcost))#

For a parametric function #f(t)=(x(t),y(t))#

derivative #(dy)/(dx)# is defined as #(dy)/(dx)=((dy)/(dt))/((dx)/(dt))#

Here #x(t)=t/sint# hence #(dx)/(dt)=(sint-tcost)/sin^2t#

and as #y(t)=cost/t^2#, #(dy)/(dt)=(-t^2sint-2tcost)/t^4#

Hence #(dy)/(dx)=-((t^2sint+2tcost)/t^4)/((sint-tcost)/sin^2t)#

= #(sin^2t(t^2sint+2tcost))/(t^4(sint-tcost))#

= #(sin^2t(tsint+2cost))/(t^3(sint-tcost))#