Final answer:
To find dy/dx and d2y/dx2 for the given equations x = et and y = te-t, we can differentiate y with respect to x using the chain rule.
Step-by-step explanation:
To find dy/dx and d2y/dx2 for the given equations x = et and y = te-t, we can differentiate y with respect to x using the chain rule.
First, we find dy/dt by differentiating y = te-t with respect to t. dy/dt = 1 * e-t - te-t * (-1 * e-t) = e-t + te-t.
Then, we find dx/dt by differentiating x = et with respect to t. dx/dt = e-t * (-1) = -e-t.
Finally, we can find dy/dx by dividing dy/dt by dx/dt. dy/dx = (e-t + te-t) / (-e-t) = -1 - t.
To find d2y/dx2, we differentiate dy/dx = -1 - t with respect to x. d2y/dx2 = d/dx(-1 - t) = 0 - 1 = -1.