If x= t, t>0 and y = 1 t then d ^2 y d x^2 =

Step-by-step solution

Given the parametric equations x = t and y = 1 t , the first derivative is determined using the chain rule: d y d x = d y d t d x d t = - 1 t^2 1 t = - 1 t For the second derivative, d ^2 y d x^2 = d d t ( d y d x ) d t d x . Differentiating d y d x = - 1 t with respect to t yields 1 t^2 , and since d x d t = 1 t , it follows that d t d x = t . Consequently: d ^2 y d x^2 = 1 t^2 t = 1 t Observing that 1 t = - ( - 1 t ) = - d y d x , the result simplifies to d ^2 y d x^2 = - d y d x , which corresponds to option B .