List of top Mathematics Questions on Differentiability asked in WBJEE

Let $F(x) = e^x, G(x) = e^{-x}$ and $H(x) = G(F(x))$, where $x$ is a real variable. Then $\frac{dH}{dx}$ at $x = 0$ is

For all real values of \(a_0, a_1, a_2, a_3\) satisfying \(a_{0}+\frac{a_{1}}{2}+\frac{a_{2}}{3}+\frac{a_{3}}{4}=0\), the equation \(a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}=0\) has a real root in the interval

Rolle's theorem is applicable in the interval $[-2, 2]$ for the function

Let $f(x) = \tan^{-1}x$. Then $f'(x) + f''(x)$ is = $0$, when $x$ is equal to