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Question
Differentiate the following w.r.t. x : `root(3)((4x - 1)/((2x + 3)(5 - 2x)^2)`
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Solution
Let y = `root(3)((4x - 1)/((2x + 3)(5 - 2x)^2)`
Then log y = `log[(4x - 1)/((2x + 3)(5 - 2x)^2)]^(1/3)`
= `(1)/(3)log[(4x - 1)/((2x + 3)(5 - 2x)^2)]`
= `(1)/(3)[log(4x - 1) - log(2x + 3)(5 - 2x)^2]`
= `(1)/(3)log(4x - 1) - (1)/(3)log(2x + 3) - (2)/(3)log(5 - 2x)`
Differentiating both sides w.r.t. x, we get
`(1)/y."dy"/"dx" = (1)/(3)"d"/"dx"[log(4x - 1)] - (1)/(3)"d"/"dx"[log(2x + 3)] - (2)/(3)"d"/"dx"[log(5 - 2x)]`
= `(1)/(3) xx (1)/(4x - 1)."d"/"dx"(4x - 1) - (1)/(3) xx (1)/(2x + 3)."d"/"dx"(2x + 3) - (2)/(3) xx (1)/(5 - 2x)."d"/"dx"(5 - 2x)`
= `(1)/(3(4x - 1)). (4 xx 1 - 0) - (1)/(3(2x + 3)).(2 xx 1 + 0) - (2)/(3(5 - 2x)).(0 - 2 xx 1)`
∴ `"dy"/"dx" = y[(4)/(3(4x - 1)) - (2)/(3(2x + 3)) + (4)/(3(5 - 2x))]`
= `root(3)((4x - 1)/((2x + 3)(5 - 2x)^2))[(4)/(3(4x - 1)) - (2)/(3(2x + 3)) + (4)/(3(5 - 2x))]`.
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