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.

Options

A. `(du)/(dx) uw - u (dv)/(dx) w + uv(dw)/(dx)`
B. `-(du)/(dx) uw + u v (dw)/(dx) + u(dw)/(dx)w`
C. `(du)/(dx) vw + u (dw)/(dx)w + uv(dw)/(dx)`
D. None of the above

Answer

C

Explanation

Let ` y = uvw rArr log y = log (uvw)`
` rArr log y= log u + log v + log w ` (i)
On differentiating both sides w.r.t.x we get
`(1)/(y) (dy)/(dx) = (1)/(u) (du)/(dx) + (1)/(v) (dv)/(dx) + (1)/(w) (dw)/(dx)`
`rArr (dy)/(dx) = y {(1)/(u) (du)/(dx) + (1)/(v) (dv)/(dx) + (1)/(w) (dw)/(dx)}`
`= (uvw){(1)/(u) (du)/(dx) + (1)/(v) (dv)/(dx) + (1)/(w) (dw)/(dx)} " "` [from Eq. (i) ]
` = uv (dw)/(dx) + uw (dv)/(dx) + wv(du)/(dv) ` ....(ii).