If u, v and w are functions of x, then show that `d/(dx)(udotvdotw)=(d u)/(dx)vdotw+udot(d v)/(dx)dotw+udotv(d w)/(dx)` in two ways - first by repeated application of product rule, second by logarithmic differentiation.

Explanation

Using Pro(du)ct rule, in which u and v are taken as one


`(d(uvw))/((dx))​=(d(uv))/((dx))​w+(d(w)​uv)/((dx))`

=`w.v.(d(u))/((dx))​+(d(v))/((dx))​u.w+(d(w)​uv)/((dx))`


Now using logarithmic,

y=uvw

taking log on both sides, we have

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