QuestionFind the $1^{\text {st }}$ and $2^{\text {nd }}$ derivative of y with respect to x from the given parametric equations a. $\quad x=\ln \left(\frac{1+e^{-2 t}}{1-e^{-2 t}}\right), y=\arctan (2 t)$

Studdy Solution
Calculate the second derivative $\frac{d^2y}{dx^2}$:
$$\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}$$
Substitute $\frac{d}{dt}\left(\frac{dy}{dx}\right)$ and $\frac{dx}{dt}$ from previous steps.
The first derivative is $\frac{dy}{dx} = \frac{-1(1+e^{-2t})(1-e^{-2t})}{2e^{-2t}(1+4t^2)}$.
The second derivative $\frac{d^2y}{dx^2}$ involves further simplification and substitution, which can be computed following the outlined steps.