Angular momentum - Question on Zare's use of the Vector Model to Estimate Probability

My question regards a specific example, although I get the feeling that the answer is simpler than I imagine. In Zare's *Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics pgs 53-54*, He discusses the vector model for coupled and uncoupled representations for the addition of two angular momenta. 


I've scanned and included the picture I am going to refer to for clarity:


[![Fig. 1](https://i.stack.imgur.com/JqvEJ.png)](https://i.stack.imgur.com/JqvEJ.png)
The image above is a modified representation of the ***uncoupled representation***, although it appears very similar to the coupled diagram, the difference is that **j1** has been arbitrarily fixed, and **j2** is precessing in the dashed circle, around an axis parallel to z.


Zare expands the dot product of $\mathbf{j\_1}$ and $\mathbf{j\_2}$
(Eq 2.34):


$\mathbf{j\_1 . j\_2} = |\mathbf{j\_1}||\mathbf{j\_2}| cos(\gamma)$


$=|\mathbf{j\_1}||\mathbf{j\_2}| [ cos(\theta\_1) cos(\theta\_2)+ sin(\theta\_1) sin(\theta\_2) cos(\theta\_1) cos(\phi\_1 -\phi\_2)]$


So far, so good, no problem. Then he instructs: 


"Let us differentiate Eq (2.34) with respect to time to obtain the rate of change of the length of $\mathbf{j}$. From the law of cosines


$|\mathbf{j}|^2 = |\mathbf{j\_1}|^2 + |\mathbf{j\_2}|^2 - 2 |\mathbf{j\_1}| |\mathbf{j\_2}| cos(\gamma)$


$=|\mathbf{j\_1}|^2 + |\mathbf{j\_2}|^2 - 2 \mathbf{j\_1} . \mathbf{j\_2}$


So that


$\frac d{dt} (\mathbf{j\_1} . \mathbf{j\_2}) = -\frac 12 \frac d{dt} [ |\mathbf{j}|^2 + |\mathbf{j\_1}|^2 - |\mathbf{j\_2}|^2 ]$"


And **THIS** next step is where I get lost:


He ***IMMEDIATELY*** equates that expression with the following:


$$ = -|\mathbf{j}| \frac {d|\mathbf{j}|}{dt} $$


So that there's no confusion, what I do not understand is the statement:


$$\frac d{dt} (\mathbf{j\_1} . \mathbf{j\_2}) = -\frac 12 \frac d{dt} [ |\mathbf{j}|^2 + |\mathbf{j\_1}|^2 - |\mathbf{j\_2}|^2 ] = -|\mathbf{j}| \frac {d|\mathbf{j}|}{dt}$$




---


Now, if I explicitly expand all components of the vector norms: $\mathbf{j\_1}$,$\mathbf{j\_2}$,and $\mathbf{j}$, and try to solve for what's inside the brackets on the LHS, I just regenerate the expression: 


$|\mathbf{j}|^2 =|\mathbf{j\_1}|^2 + |\mathbf{j\_2}|^2 - 2 \mathbf{j\_1} . \mathbf{j\_2}$, which is of no use in aiding me as far as I can see.


Furthermore, it is explicitly stated that the only time-dependent component of this system is the dihedral angle $\phi\_1 - \phi\_2$, and in any expansion of the vector $|\mathbf{j}|$ into its components, a time derivative would render the any other independent parts mute. 


I am thoroughly confused. I have consulted the following textbooks and not found any sign of this exact derivation:


*Cohen-Tannoudji (volumes I and II), Feynman's lectures on Physics Vol III, Shankar, Atkin's physical chemistry, Engel and Reid's physical chemistry, Gaziorowicz, and Eisberg and Resnick.*


What I suspect is that I'm simply lacking in some fundamental piece of calculus or arithmetic. Whether there's some simple trick of Vector Calculus that I haven't employed to give me the identity placed here? I would love a direct answer, but will also accept a nudge in the right direction. Right now it just feels like I'm leafing through books without a compass.


Best, 


and ***THANKS!***




As mentioned in the comment by Weijun Zhou, the length of the vectors $\mathbf{j}\_1$ and $\mathbf{j}\_2$ does not change as $\mathbf{j}\_2$ precesses about $\mathbf{j}\_1$, only the total vector $\mathbf{j}$ changes its length. Knowing this just requires application of the sum and chain rule of derivations that is


$
-\frac{1}{2}\frac{d}{dt}|\mathbf{j}|^2=-2|\mathbf{j}|\frac{1}{2}\frac{d}{dt}|\mathbf{j}|
$


and the time derivatives of $\mathbf{j}\_1$ and $\mathbf{j}\_2$ are zero.


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