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 \(x=3\). 
 

If $y = a \cos(\log x) + b \sin(\log x)$, then $x^2y'' + xy'1$ is:
(a) Differentiate $\sqrt{e^{\sqrt{2x}}}$ with respect to $e^{\sqrt{2x}}$ for $x>0$.
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 $\frac{dy}{dx}$.
Assertion (A): \( f(x) = [x] \), \( x \in \mathbb{R} \), the greatest integer function is not differentiable at \( x = 2 \). Reason (R): The greatest integer function is not continuous at any integral value.
Function f is defined as: \[ f(x) = \begin{cases} 2x + 2, & x<2 \\ k, & x = 2 \\ 3x, & x>2. \end{cases} \text{Find the value of } k \text{ for which } f \text{ is continuous at } x = 2. \]
Prove that the greatest integer function defined by f(x)=[x],0<x<3 is not differentiable at x=1 and x=2.
If \(e^y(x+1)=1\),show that \(\frac{d^2y}{dx^2}=(\frac{dy}{dx})^2\)

If (x-a)2+(y-b)2=c2, for some c>0 prove that
[1+(\(\frac{dy}{dx}\))2]\(^{\frac{3}{2}}\)/\(\frac{d^2y}{dx^2}\) is a constant independent of a and b

Differentiate the following w.r.t. \(x\):
\(cosx.cos2x.cos3x\)
Differentiate w.r.t. x the function:\(x^x+x^a+a^x+a^a\), for some fixed \(a>0\) and \(x>0\)
Check the differentiability of the function \( f(x) = \lfloor x \rfloor \) at \( x = -3 \), where \( \lfloor \cdot \rfloor \) denotes the greatest integer function.
Which of the following statements is true for the function:

\[ f(x) = \begin{cases} x^2 + 3, & \text{if } x \neq 0, \\ 1, & \text{if } x = 0. \end{cases} \]

Let \( f(x) \) be a continuous function on \([a, b]\) and differentiable on \((a, b)\). Then, this function \( f(x) \) is strictly increasing in \((a, b)\) if:
\(\text {Find} \ \frac {dy}{dx}:\) \(y=sin^{-1}(\frac {2x}{1+x^2})\)
Find \( \frac {dy}{dx}:\) \(sin^2y+cos\ xy=\pi\)

Find \(\frac{dy}{dx}\),if y=12(1-cost),x=10(t-sint),\(-\frac{\pi}{2}\)<t<\(\frac{\pi}{2}\)

Find \(\frac {dy}{dx}\):  \(ax+by^2=cos\ y\)
Find \(\frac {dy}{dx}\): \(2x+3y=sin\ x\)
Find \(\frac {dy}{dx}\): \(xy+y^2=tan\ x+y\)