Question

How do you derive `y = (x^2+8x+3)/x^(1/2)` using the quotient rule?

Answer 1

`(3x^2 + 8x - 3)/(2x^(3/2))`

Explanation:

`=[(x^2 + 8x + 3)'(x^(1/2)) - (x^(1/2))'(x^2 +8x + 3)]/(x^(1/2))^2`
`=[x^(1/2)(2x+8) - 1/2x^(-1/2)(x^2+8x+3)]/x`
`=[2x^(3/2) + 8x^(1/2) - 1/2x^(3/2) - 4x^(1/2) - 3/2x^(-1/2)]/x`
`=[3/2x^(3/2) + 4x^(1/2)-3/2x^(-1/2)]/x`
`=[x^(-1/2)[3/2x^2 + 4x -3/2]]/x`
`=[(3x^2 + 8x + 3)/2]/x^(3/2)`
`=(3x^2 + 8x + 3)/(2x^(3/2))`