Difficult derivative involving vectors and trig

[Tornam]

Hello,

I am working through some very old (1980's) computer code and need to understand how a particular derivative was calculated. Can someone explain to me how it is that if:

\vec{a}=\vec{f}\times\vec{g}
\vec{b}=\vec{h}\times\vec{g}

and

sin(\phi)=\frac{\left|\vec{a}\times\vec{b}\right|}{\left|a\right|\left|b\right|}
cos(\phi)=\frac{\vec{a}\cdot\vec{b}}{\left|a\right|\left|b\right|}

then:

\frac{d\phi}{d\vec{f}} = -\frac{\left|g\right|}{\left|a\right|^2}\cdot\vec{a}?

I would very much appreciate any help with this!

Thanks :)

 

[haruspex]

Have you tried differentiating sin phi wrt f?