We just learned derivatives and now on the Chain Rule.

Question

y = r/[(sqrt)(r(sqrd)+1)]

y' = [[(sqrt)(r(sqrd)+1)(r)'] - [(r)(sqrt)(r(sqrd)+1)']] / [(sqrt)(r(sqrd)+1)](sqrd)
y' = (sqrt)(r(sqrd)+1) - r[1/2(r(sqrd)+1)(raised to -1/2)](2r)

My final answer is -3r/[2(sqrt)(r(sqrd)+1)]. However, my textbook shows [(r(sqrd)+1)(raised to -3/2)]

I have a feeling I am combining the Quotient Rule and Chain Rule into one step. Is this the mistake I am making?

Answer 1

If I read your weird syntax correctly, you have

y = r/√(r^2+1)

Using the quotient rule, if u and v are functions of r,

y = u/v
y' = (u'v-uv')/v^2
We have u=r and v=√(r^2+1)
u' = 1 and v' = r/√(r^2+1)

y' = [(1)√(r^2+1)-r(r/√(r^2+1))]/√(r^2+1)^2
= (√(r^2+1)-r^2/√(r^2+1)]/(r^2+1)
= (r^2+1-r^2)/(r^2+1)^(3/2)
= 1/(r^2+1)^(3/2)