91Ó°ÊÓ

We have the expressions for \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) as: \(\frac{dx}{dt} = \cos t - t\sin t\) \(\frac{dy}{dt} = 1 + \cos t\) We will use the Chain Rule to find \(\frac{dx}{dy}\): \(\frac{dx}{dy} = \frac{\frac{dx}{dt}}{\frac{dy}{dt}}\) Now substitute the expressions given: \(\frac{dx}{dy} = \frac{\cos t - t \sin t}{1 + \cos t}\)

Now, we need to find the second derivative of x with respect to y. We take the derivative of the first derivative with respect to t and then divide by the derivative of y with respect to t: \(\frac{d^2x}{dy^2} = \frac{\frac{d(\frac{dx}{dy})}{dt}}{\frac{dy}{dt}}\) Take the derivative of \(\frac{dx}{dy}\) with respect to t: \(\frac{d(\frac{dx}{dy})}{dt} = \frac{(1+\cos t)(-t\cos t - 2\sin t) - (\cos t - t\sin t)\sin t}{(1+\cos t)^2}\) Now, divide by the expression for \(\frac{dy}{dt}\): \(\frac{d^2x}{dy^2} = \frac{(1+\cos t)(-t\cos t - 2\sin t) - (\cos t - t\sin t)\sin t}{(1+\cos t)^3}\) Simplify the expression: \(\frac{d^2x}{dy^2} = -2 - \frac{\pi}{2}\) Finally, obtain the expression for the second derivative of x with respect to y: \(\frac{d^2x}{dy^2} = -\frac{(\pi + 4)}{2}\)