If the function f is continuous at $x = 1$, where $f(x) = \dfrac{1 + \cos(\pi x)}{\pi(1-x)^2}$, for $x \neq 1$, then the value of $f(1)$ is....

If $y = \log x^x$, then the value of $\left(\dfrac{dy}{dx}\right)^2$ at $x = e^2$ is

If $y = \cos^{-1}(\sin x)$, where $\dfrac{\pi}{2} < x < \pi$, then $\dfrac{dy}{dx} = ...$

If $y = \left(\dfrac{ax+b}{cx+d}\right)$, then $2\dfrac{dy}{dx} \cdot \dfrac{d^3 y}{dx^3}$ is equal to