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Differentiate the following w.r.t.x: (3x-5-13x-5)5 - Mathematics and Statistics

Sum

Differentiate the following w.r.t.x: `(sqrt(3x - 5) - 1/sqrt(3x - 5))^5`

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Solution

Let y = `(sqrt(3x - 5) - 1/sqrt(3x - 5))^5`

Differentiating w.r.t. x,we get,
`"dy"/"dx" = "d"/"dx"(sqrt(3x - 5) - 1/sqrt(3x - 5))^5`

`= 5(sqrt(3x - 5) - 1/sqrt(3x - 5))^4."d"/"dx"(sqrt(3x - 5) - 1/sqrt(3x - 5))`

`= 5(sqrt(3x - 5) - 1/sqrt(3x - 5))^4.["d"/"dx"(3x - 5)^(1/2) - "d"/"dx"(3x - 5)^(-1/2))]`

`= 5(sqrt(3x - 5) - 1/sqrt(3x - 5))^4 xx [1/2(3x - 5)^(-1/2)."d"/"dx"(3x - 5) - (-1/2)(3x - 5)^(-3/2)."d"/"dx"(3x - 5)]`

`= 5(sqrt(3x - 5) - 1/sqrt(3x - 5))^4 xx [1/(2sqrt(3x - 5)).(3 xx 1 - 0) + 1/(2(3x - 5)^(3/2)).(3 xx 1 - 0)]`

`= 15/2(sqrt(3x - 5) - 1/sqrt(3x - 5))^4.[3/(2sqrt(3x - 5)) +3/(2(3x - 5)^(3/2))]`

`= 15/2(sqrt(3x - 5) - 1/sqrt(3x - 5))^4.[(3x - 5 + 1)/(3x - 5)^(3/2)]`

`= (15(3x - 4))/(2(3x - 5)^(3/2))(sqrt(3x - 5) - 1/sqrt(3x - 5))^4`.

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