Question

So, I have a homework question on The Quotient Rule (taking derivitives) and I would like to know if I did it right.
Calculate: y=(1/sqrt(x))-(1/(5th root of x^3). My 1st step was to change the sqrt to x^1/2, and the 5th root to x^1/5+^3. So then I got
y=(1/x^1/2)-(1/x^16/5). I subtracted the fractions to get 1 fraction
y=1/x^-27/10. I then used the Quotient Rule (y=(u'v-v'u)/v^2)) to derive
y'=(0(x^-27/10)-(-27/10x^-37/10)(1))/(x^(-27/10))^2.
My final answer looks like
((27/10)x^(-37/10))/(x^(-27/10))^2.
I am hoping I did this right, any input would be helpful.

Answer 1

5th root of x^3 is (x^3)^1/5 = x^(3*1/5), not x^(3+1/5)

So, you start out with

y = x^(-1/2) - x^(-3/5)
Now just use the power rule, to get
y' = (-1/2)x^(-3/2) - (-3/5)x^(-8/5)

If you insist on combining the fractions, then you wind up with

6 - 5x^(1/10)
--------------------
10x^(8/5)