For $x > y > 0$, if $x^5 y^6 = (x + y)^{11}$, then $\frac{d^2y}{dx^2}$ is

Let a function $f(x)=\alpha+(\beta^2+5\beta+6)|x|+\gamma|x|^4$, where $\alpha, \beta$ and $\gamma$ are real number. Then the function $f(x)$ is differentiable at $x=0$ if

If $\mathrm{e}^{\mathrm{y}}=\mathrm{x}^{\mathrm{x}}$, then which of the following is true ?

If $y = \left(x + \sqrt{x^2+1}\right)^m$, then $\frac{dy}{dx}$ is

If $y = x\log_e\left(\dfrac{x}{2+x}\right)$, then $x^3\dfrac{d^2y}{dx^2}$ is equal to:

If $y = \left[\dfrac{2}{1+x^{b-a}+x^{c-a}} + \dfrac{2}{1+x^{c-b}+x^{a-b}} + \dfrac{2}{1+x^{a-c}+x^{b-c}}\right]^x$, then $\dfrac{dy}{dx}$ is equal to :

Match List-I with List-II

Choose the correct answer from the options given below: