Which of the expressions below are equivalent to u'vw +uv'w - uvw'/ w2 if u,v and w are all functions of x and a dash denotes a derivative with respect to x.

Question

Which of the expressions below are equivalent to u'vw +uv'w - uvw'/ w2 if u,v and w are all functions of x and a dash denotes a derivative with respect to x.

d/dx (uv/w)

d/dx (u/vw)

d/dx (1/uvw)

d/dx (uvw)

Answer 1

Let's do these one at a time, starting with the first one.

d/dx (uv/w)

Let's first use quotient rule: as shown below:

Let 

g(x) = uv

h(x) = w

g'(x) = u'v + uv' (Product rule)

h'(x) = w'

f'(x) = (u'vw + uv'w - uvw')/w²

Since that is the answer, no need to go through the others.