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

Let $h(x) = \frac{5(f(x))^3}{3} + \frac{(f(x))^2}{2} + 2f(x) + 100$,where $f(x)$ is a differentiable function. Then which one of the following is correct?

Let $f(x)$ be a polynomial function such that $f(x)+f^{\prime}(x)+f^{\prime \prime}(x)=x^{5}+64$. Then,the value of $\lim _{x \rightarrow 1} \frac{f(x)}{x-1}$ is:

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

If $y=\frac{\cos x}{1+\sin x}$,then
$(a)$ $\frac{dy}{dx}=\frac{-1}{1+\sin x}$
$(b)$ $\frac{dy}{dx}=\frac{1}{1+\sin x}$
$(c)$ $\frac{dy}{dx}=-\frac{1}{2} \sec ^2\left(\frac{\pi}{4}-\frac{x}{2}\right)$
$(d)$ $\frac{dy}{dx}=\frac{1}{2} \sec ^2\left(\frac{\pi}{4}-\frac{x}{2}\right)$

If $f(t) = \frac{1 + \operatorname{cosec} t}{1 - \operatorname{cosec} t}$ for $0 < t < \frac{\pi}{2}$ and $f^{\prime}(t) = f(t) g(t)$,then $g(t) =$