Determine a formula for the product rule in the case where we are asked to differentiate the product of four functions. That is if y=m(x)p(x)q(x)r(x), determine a formula for dy/dx.

just use the product rule

y = mpqr
y' = m'(pqr) + m(pqr)'
Now use the product rule again to get
y' = m'(pqr) + m(p'(qr)+p(qr)')
and again to end up with
y' = m'(pqr)+p'(mqr)+q'(mpr)+r'(mpq)