If $y = \sin^{-1}\left(\dfrac{25 - x^2}{25 + x^2}\right)$, then $y'(1)$ is...

If $f(x)$ and $g(x)$ are inverse functions of each other and $f(x) = x + e^x$, then $g'(x) = $...

If $3y^2 - 2xy - x = 0$, then the value of $\dfrac{dy}{dx}$ at $y = 2$ is...

Rolle's theorem holds for monic quadratic polynomial $f(x)$ on the interval $[\alpha, \alpha + 3]$ where $f(\alpha) = 0$. Similarly, $g(x) = f(x) + 2$ also follows Rolle's theorem in the interval $[\beta, 3]$ where $g(3) = 0$, such that the value of $c$ is the same for both $f(x)$ and $g(x)$. Then the value of $(f \circ g)(\alpha)$ is...