Find the derivative of the function: $4 \sqrt{x} - 2$. (Assume $a, b, c, d, p, q, r, s$ are fixed non-zero constants and $m, n$ are integers.)

Find the derivative: $\frac{d}{dx} \left[ \frac{e^{ax}}{\sin(bx + c)} \right]$

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

Let $f: \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow \mathbb{R}$ be a differentiable function such that $f(0)=\frac{1}{2}$. If $\lim _{x \rightarrow 0} \frac{x \int_0^x f(t) dt}{e^{x^2}-1}=\alpha$,then $8 \alpha^2$ is equal to:

Differentiate the following with respect to $x$: $e^{x^{3}}$

$\frac{d}{dx}(\sin(x^2) + \cos(x^2)) = $ . . . . . .