$\frac{d}{dx} \log |x| = ......, (x \ne 0)$

Find the derivative: $\frac{d}{dx}(x e^{x^2}) = $

If $y = \frac{ax+b}{cx+d}$ and $\frac{dx}{dy} = \frac{ad-bc}{Py^2+Qy+R}$,then $P+Q+R =$

If $f(x) = x + 2,$ then $f'(f(x))$ at $x = 4$ is

For the function $f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + \dots + \frac{x^2}{2} + x + 1$,prove that $f^{\prime}(1) = 100 f^{\prime}(0)$.

Let $f(x) = x^m$, where $m$ is a non-negative integer. The value of $m$ such that the equality $f^{\prime}(a+b) = f^{\prime}(a) + f^{\prime}(b)$ is valid for all $a, b > 0$ is