how find derivative of xy3/(1+y2)

Question

find the derivative of x times y cubed divided by (1 + y squared)

Answer 1

xy³/(1+y²).

Let u=xy³ and v=(1+y²)^-1.

u'=3xy²y'+y³; v'=-(1+y²)^-2(2yy'), where y'=dy/dx.

d(uv)/dx=vu'+uv'=(3xy²y'+y³)/(1+y²)-xy³(1+y²)^-2(2yy')=(3xy²y'+y³)/(1+y²)-2xy⁴y'/(1+y²)².

If z=uv, then z'=(3xy²y'+y³)/(1+y²)-2xy⁴y'/(1+y²)² is the differential of the given expression.

Since we are not given the relationship between y and x, this differential has to include the unknown y'.