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Check the differentiability of function $f(x)=x|x|$ at $x=0$.

Find k so that $f(x)=\begin{cases}\frac{x^{2}-2x-3}{x+1},&x\ne-1\\ k,&x=-1\end{cases}$ is continuous at $x=-1$.

Check the differentiability of f(x) at $x=-2$ if $f(x)=\begin{cases}2x-3,-3\le x\le-2\\ x+1,-2<x\le0\end{cases}$.

Check for differentiability of the function f defined by $f(x)=|x-5|$, at the point $x=5$.

Verify whether the function f defined by $f(x)=\begin{cases}x~sin(\frac{1}{x}),x\ne0\\ 0&,x=0\end{cases}$ is continuous at $x=0$ or not.

Check whether the function $f(x)=x^{2}|x|$ is differentiable at $x=0$ or not.

Find the value of a and b so that function f defined as : $$ f(x) = \begin{cases} \frac{x-2}{|x-2|} + a, & \text{if } x < 2 \\ a+b, & \text{if } x = 2 \\ \frac{x-2}{|x-2|} + b, & \text{if } x > 2 \end{cases} $$ is a continuous function.

Check the differentiability of $f(x)=\begin{cases}x^{2}+1,&0\le x<1\\ 3-x,&1\le x\le2\end{cases}$ at $x=1.$

Assertion (A): $f(x) = \begin{cases} x\sin\frac{1}{x}, & x\neq 0 \\ 0, & x=0 \end{cases}$ is continuous at $x=0$.

If $x=a\sin 2t, y=a(\cos 2t+\log\tan t)$ then find $\frac{dy}{dx}$

If $y=x^{\frac{1}{x}}$ then find $\frac{dy}{dx}$ at $x=1$.

22. If $(x^{2}+y^{2})^{2}=xy$, then find $\frac{dy}{dx}$

If $y=(x+\sqrt{x^{2}-1})^{2}$;, then show that $(x^{2}-1)(\frac{dy}{dx})^{2}=4y^{2}.$

(a) Differentiate $\text{sec}^{-1}\left(\frac{1}{\sqrt{1-x^2}}\right)$ w.r.t. $\sin^{-1}\left(2x\sqrt{1-x^2}\right)$.
OR
(b) If $y = \tan x + \sec x$, then prove that $\frac{d^2y}{dx^2} = \frac{\cos x}{(1-\sin x)^2}$.

(a) If $f(x) = \begin{cases} x^2, & \text{if } x \geq 1 \\ x, & \text{if } x < 1 \end{cases}$, then show that $f$ is not differentiable at $x=1$.
OR
(b) Find the value(s) of '$\lambda$', if the function $f(x) = \begin{cases} \frac{\sin^2 \lambda x}{x^2} & \text{if } x \neq 0 \\ 1 & \text{if } x=0 \end{cases}$ is continuous at $x=0$.

If $y=\frac{\cos x-\sin x}{\cos x+\sin x}$ then $\frac{dy}{dx}$ is:

The value of k for which function $f(x)=\begin{cases}x^{2},&x\ge0\\ kx,&x<0\end{cases}$ is differentiable at $x=0$ is:

7. If $y=\sin^{2}(x^{3})$, then $\frac{dy}{dx}$ is equal to :

6. The function $f(x)=|x|$ is

The value of k for which$ f(x)=\begin{cases}3x+5,&x\ge2\\ kx^{2},&x<2\end{cases}$ is a continuous function, is :