List of top Mathematics Questions on Continuity and differentiability asked in CBSE CLASS XII

Sports car racing is a form of motorsport which uses sports car prototypes. The competition is held on special tracks designed in various shapes. The equation of one such track is given as 

(i) Find \(f'(x)\) for \(0<x>3\). 
(ii) Find \(f'(4)\). 
(iii)(a) Test for continuity of \(f(x)\) at \(x=3\). 
OR 
(iii)(b) Test for differentiability of \(f(x)\) at \(x=3\). 
 

(a) Differentiate $\sqrt{e^{\sqrt{2x}}}$ with respect to $e^{\sqrt{2x}}$ for $x>0$.
If $y = a \cos(\log x) + b \sin(\log x)$, then $x^2y'' + xy'1$ is:
If \[ f(x) = \begin{cases} \frac{\log(1 + ax) + \log(1 - bx)}{x}, & \text{for } x \neq 0 \\ k, & \text{for } x = 0 \end{cases} \] is continuous at $x = 0$, then the value of $k$ is:
If $\tan^{-1} (x^2 - y^2) = a$, where $a$ is a constant, then $\frac{dy}{dx}$ is:
Let $f(x) = |x|$, $x \in \mathbb{R}$. Then, which of the following statements is incorrect?
Differentiate $y = \cos^{-1}\left( \frac{1 - x^2}{1 + x^2} \right)$ with respect to $x$, when $x \in (0, 1)$.
Differentiate $y = \sin^{-1}(3x - 4x^3)$ with respect to $x$, for $x \in \left[ -\frac{1}{2}, \frac{1}{2} \right]$.
If $(x)^y = (y)^x$, then find $\frac{dy}{dx}$.