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The derivation of sqrt(((x-3)(x^(2) +4)...

The derivation of `sqrt(((x-3)(x^(2) +4))/(3x^(2) + 4x + 5))` with respect to x is a .

A

`(1)/(2)sqrt(((x-3)(x^(2) +4))/(3x^(2) + 4x + 5))[(1)/(x-3) + (2x)/(x^(2) +4) -(6x +4)/(3x^(2) + 4x + 5)]`

B

`(1)/(2)sqrt(((x-3)(x^(2) +4))/(3x^(2) + 4x + 5))[(1)/(x-3) - (2x)/(x^(2) +4) +(6x +4)/(3x^(2) + 4x + 5)]`

C

`(1)/(2)sqrt(((x-3)(x^(2) +4))/(3x^(2) + 4x + 5))[(1)/(x-3) - (2x)/(x^(2) +4) -(6x +4)/(3x^(2) + 4x + 5)]`

D

None of the above

Text Solution

Verified by Experts

The correct Answer is:
A
Let `y = sqrt(((x-3)(x^(2) =4))/(3x^(2) + 4x +5))`
On taking log on both sides , we get
`log y = (1)/(2) [log (x-3) + log(x^(2) +4) - log(3x^(3) + 4x + 5)]`
On , differentiating both sides w.r.t.x, we get
`(1)/(y).(dy)/(dx) = (1)/(2) [(1)/(x-3) + (2x)/(x^(2) +4)-(6x +4)/(3x^(2) + 4x +5)]`
or ` (dy)/(dx) = (y)/(2) [(1)/(x-3) + (2x)/(x^(2) +4)-(6x +4)/(3x^(2) + 4x +5)]`
`=(1)/(2)sqrt(((x-3) (x^(2) +4))/(3x^(2) + 4x + 5) ) `
`[(1)/((x-3)) +(2x)/(x^(2) + 4)- (6x +4)/(3x^(2) + 4x +5)]`
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