If $y = x \cdot 7^x$, then the value of $\dfrac{dx}{dy}$ when $x = 1$ is

If $y = [(x+1)(2x+1)(3x+1)\ldots\ldots(nx+1)]^4$, where $n \in N$ and $\dfrac{dy}{dx}$ at $x = 0$ is $2k$, then the value of $k$ is

If $f(x) = \cos x \cos 2x \cos 4x \cos 8x \cos 16x$, then $f'\left(\dfrac{\pi}{4}\right)$ is equal to

The derivative of the function $f(x) = \cos^4 x + \sin^4 x,\ 0 \leq x \leq 2\pi$ is positive for