The graph of function $f$ contains the points $P(1, 2)$ and $Q(s, r)$. The equation of the secant line through $P$ and $Q$ is $y = \left( \frac{s^2 + 2s - 3}{s - 1} \right) x - 1 - s$. The value of $f'(1)$ is:

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 $a, b, c, d, p, q, r$ and $s$ are fixed non-zero constants and $m$ and $n$ are integers): $\frac{ax+b}{px^{2}+qx+r}$

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:

If $y = \sin(2 \sin^{-1} x)$, then $\frac{dy}{dx} = \dots$

Find the derivative of the function: $\sin x \cos x$