If $x^{x}=y^{y}$,then $\frac{d y}{d x}$ is

If $x^3 - 2x^2y^2 + 5x + y - 5 = 0$, then at $(1, 1)$, $y''_1 = $?

If $y = \sqrt{\cos x^2 + \sqrt{\cos x^2 + \sqrt{\cos x^2 + \dots \infty}}}$ and $\frac{dy}{dx} = \frac{f(x)}{2y - 1}$, then $\int f(x) dx = \dots$

If $xy = \tan^{-1}(xy) + \cot^{-1}(xy)$,then $\left(\frac{dy}{dx}\right)_{(4,2)} = ?$ (where $x, y \in \mathbb{R}$)

Consider the non-constant differentiable function $f$ of one variable which obeys the relation $\frac{f(x)}{f(y)}=f(x-y)$. If $f^{\prime}(0)=p$ and $f^{\prime}(5)=q$, then $f^{\prime}(-5)$ is

If $a(4+x^2)=x$ and $y-x^3=a^2$,then $\frac{dy}{dx}$ at $x=1$ is ...