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\[x=\frac{ y^3 }{ 6 }+\frac{ 1 }{ 2y }\]

\[\frac{ dx}{ dy }=\frac{ y^2 }{ 2 }-\frac{ 1 }{ y^2}\]

\[\int\limits_{1}^{2}\sqrt{1+((\frac{ y^2 }{ 2 })-\frac{ 1 }{ y^2 })^2} dy\]

Woops the 2 should still be there on the second term.
A -1 came down from the power rule.

u mean for 1/2y

ya

when i take derivative using quotient rule i got -1/2y^2

Do you have 2 separate problems listed? This is rather confusing....