calc question ,

I am trying to show:

g(x,y,z) = I( g(xt, yt, zt), t, 0, 1) + I( [x*g_x(xt, yt, zt) + y*g_y(xt,
yt,zt) + z*g_z(xt, yt, zt)]t, t, 0, 1),

where I( f, x, 0, 1) means integral of f wrt x from 0 to 1. The book says
to differentiate t*g(xt, yt, zt) to see this. From the product rule I got:

(tg)' = g + tg'

But where to go from here? Also how to find g'? Is g(u) = g(u(x,y)) now?


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