If x= t ^2+ t +1, y = ( t 2 )+ ( t 2 ) , then dy d x at t =1 is

Step-by-step solution

The derivative is found using parametric differentiation: dy dx = dy/dt dx/dt . Differentiating x = t^2 + t + 1 gives dx dt = 2t + 1 . For y = ( t 2 ) + ( t 2 ) , apply the chain rule: dy dt = 2 [ ( t 2 ) - ( t 2 ) ] . Combining these results, dy dx = [ ( t 2 ) - ( t 2 ) ] 2(2t + 1) . At t = 1 , dx dt = 3 and dy dt = - 2 , so dy dx = - 6 . The correct choice is D.