If $f(x) = \frac{\sin^{2} x}{1+\cot x} + \frac{\cos^{2} x}{1+\tan x}$,then $f^{\prime}\left(\frac{\pi}{4}\right)$ is

Find the derivative of the following function (it is to be understood that $a, b, c$ are fixed non-zero constants): $\frac{1}{ax^{2}+bx+c}$

Let $y = \sqrt{p x^3 y}$. If $\frac{dy}{dx} = \frac{3}{2}$ when $y = 1$, then the value of $p$ is equal to

If $y = \frac{1}{4}u^4$ and $u = \frac{2}{3}x^3 + 5$,then $\frac{dy}{dx} = $

If $y = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots$,then $\frac{dy}{dx} = $

People living on Mars,instead of the usual definition of derivative $D f(x)$,define a new kind of derivative,$D^*f(x)$ by the formula $D^*f(x) = \lim_{h \to 0} \frac{f^2(x + h) - f^2(x)}{h}$ where $f^2(x)$ means $[f(x)]^2$. If $f(x) = x \ln x$,then the value of $\left. D^*f(x) \right|_{x = e}$ is: