If $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, then $\frac{d^2y}{dx^2}$ is equal to

If $xy + \frac{x^2}{y} = x^3y + y$, then $\frac{dy}{dx}$ is equal to

Let $f(x)=\begin{cases}\dfrac{|x|}{x},&x\ne0\\1,&x=0\end{cases}$ and $g(x)=\begin{cases}x\sin\left(\dfrac{1}{x}\right),&x\ne0\\0,&x=0\end{cases}$

Then at the origin, which one of the following is true?

If $e^y(1+x)=1$, then $\dfrac{d^2y}{dx^2}$ is:

For the function $f(x) = \min\{|x|, |x-2|\}$ which of the following are TRUE?

A. Function $f(x)$ is continous on $\mathbb{R}$

B. Function $f(x)$ is differentible on $\mathbb{R}$

C. Function $f(x)$ is differentible on $\mathbb{R} \sim \{0,1,2\}$

D. Function is continous on $\mathbb{R}$ but not differentiable at $x = \dfrac{1}{2}$

Choose the correct answer from the options given below:

Derivative of $x^x$ with respect to $x\log x$ is

If $y^2 = ax + b, a,b \in \mathbb{R}$ where $\mathbb{R}$ is the set of real numbers, then $y\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2$ is equal to: