Let $\mathbb{R}$ denote the set of all real numbers. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow(0,4)$ be functions defined by

$$ f(x)=\log _e\left(x^2+2 x+4\right), \text { and } g(x)=\frac{4}{1+e^{-2 x}} $$

Define the composite function $f \circ g^{-1}$ by $\left(f \circ g^{-1}\right)(x)=f\left(g^{-1}(x)\right)$, where $g^{-1}$ is the inverse of the function $g$.

Then the value of the derivative of the composite function $f \circ g^{-1}$ at $x=2$ is ________________.

Let

$$ \alpha=\frac{1}{\sin 60^{\circ} \sin 61^{\circ}}+\frac{1}{\sin 62^{\circ} \sin 63^{\circ}}+\cdots+\frac{1}{\sin 118^{\circ} \sin 119^{\circ}} $$

Then the value of

$$ \left(\frac{\operatorname{cosec} 1^{\circ}}{\alpha}\right)^2 $$

is _____________.

If

$$ \alpha=\int\limits_{\frac{1}{2}}^2 \frac{\tan ^{-1} x}{2 x^2-3 x+2} d x $$

then the value of $\sqrt{7} \tan \left(\frac{2 \alpha \sqrt{7}}{\pi}\right)$ is _________.

(Here, the inverse trigonometric function $\tan ^{-1} x$ assumes values in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.)

A temperature difference can generate e.m.f. in some materials. Let S be the e.m.f. produced per unit temperature difference between the ends of a wire, σ the electrical conductivity and κ the thermal conductivity of the material of the wire. Taking M, L, T, I and K as dimensions of mass, length, time, current and temperature, respectively, the dimensional formula of the quantity $Z = \frac{S^2 \sigma}{\kappa}$ is :