If $y = [(x+1)(2x+1)(3x+1) \ldots (nx+1)]^4$,then $\frac{dy}{dx}$ at $x=0$ is

If $y = x^x + x^{\frac{1}{x}}$,then $\frac{dy}{dx}$ is equal to

If $y = (\tan x)^{\cot x}$,then $\frac{dy}{dx} =$

If $u, v$ and $w$ are functions of $x,$ then show that $\frac{d}{d x}(u \cdot v \cdot w) = \frac{d u}{d x} \cdot v \cdot w + u \cdot \frac{d v}{d x} \cdot w + u \cdot v \cdot \frac{d w}{d x}$ in two ways - first by repeated application of product rule,second by logarithmic differentiation.

If $y = \tan x \tan 2x \tan 3x$,then $\frac{dy}{dx}$ is equal to:

$\text{If } \frac{d}{dx} \left( \frac{x^2+1}{(x^2+5)(x^2+9)} \right) = \frac{2x(x^2+1)}{(x^2+5)(x^2+9)} \left[ \frac{1}{f(x)} - \frac{1}{g(x)} - \frac{1}{h(x)} \right], \text{ then } 2h(x) - f(x) - g(x) = $