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Question
Differentiate the function with respect to x.
`sqrt(((x-1)(x-2))/((x-3)(x-4)(x-5)))`
Solution
Let, y = `sqrt(((x - 1)(x - 2))/((x - 3)(x - 4)(x - 5)))` ...(1)
or, y = `[((x - 1)(x - 2))/((x - 3)(x - 4)(x - 5))]^(1/2)`
Taking logarithm of both sides,
`log y = 1/2 ((x - 1)(x - 2))/((x - 3)(x - 4)(x - 5)) ... [because log m^n = n log m]`
या `log y = 1/2 log (x - 1)(x - 2) - 1/2 log (x - 3)(x - 4)(x - 5) ... [because log m/n = log m - log n]`
`= 1/2 [log (x- 1) + log (x - 2)] - 1/2 [log (x - 3) + log (x - 4) + log (x - 5)] ...[because log m . n = log m + log n]`
Representing both sides by x,
`1/ y dy/dx = 1/2 [d/dx log (x - 1) + d/dx log (x - 2)] - 1/2 [d/dx log (x - 3) + d/dx log (x - 4) + d/dx log (x - 5)]`
`= 1/2 y [1/(x - 1) d/dx (x - 1) + 1/(x - 2) d/dx (x - 2)] - 1/2 y [1/(x - 3) d/dx (x - 3) + 1/(x - 4) d/dx (x - 4) + 1/(x - 5) d/dx (x - 5)]`
`= 1/2 y [1/(x - 1) + 1/(x - 2)] - 1/2 y [1/(x - 3) + 1/(x - 4) + 1/(x - 5)]`
`= 1/2 y [1/(x - 1) + 1/(x - 2) - 1/(x - 3) - 1/(x - 4) - 1/(x - 5)]`
Putting the value of y from equation (1),
`dy/dx = 1/2 sqrt(((x - 1)(x - 2))/((x - 3)(x - 4)(x - 5))) [1/(x - 1) + 1/(x - 2) - 1/(x - 3) - 1/(x - 4) - 1/(x - 5)]`
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