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Find the derivative of $\tan^{-1}\!\left(\dfrac{\sin x-\cos x}{\sin x+\cos x}\right)$ with respect to $\dfrac{x}{2}$, for $x\in\left(0,\dfrac{\pi}{2}\right)$.
$\dfrac{\sin x-\cos x}{\sin x+\cos x} = \tan\!\left(x-\dfrac{\pi}{4}\right)$
So $f(x)=\tan^{-1}\!\left(\tan\!\left(x-\dfrac{\pi}{4}\right)\right)=x-\dfrac{\pi}{4}$ (within the principal range).
$\dfrac{df}{dx}=1$.
Let $g(x)=x/2$, so $dg/dx=1/2$.
Derivative of $f$ w.r.t. $g$: $\dfrac{df/dx}{dg/dx}=\dfrac{1}{1/2}=\mathbf{2}$
Stable equilibrium: magnetic moment $\vec{m}$ is parallel to $\vec{B}$ (minimum energy, $U = -mB$).
Unstable equilibrium: $\vec{m}$ is antiparallel to $\vec{B}$ (maximum energy, $U = +mB$).
From the figures: In orientation (b), $\vec{m}$ is along $+z$ (parallel to $\vec{B}$) $\Rightarrow$ stable. In orientation (d), $\vec{m}$ is along $-z$ (antiparallel to $\vec{B}$) $\Rightarrow$ unstable. Answer: (b) and (d).
The 'Magna Carta' (1215) is significant in the UK for:
The Magna Carta established the principle that everyone, including the king, is subject to the law.
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