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प्रश्न
Differentiate the following with respect to x.
`sqrt(((x - 1)(x - 2))/((x - 3)(x^2 + x + 1)))`
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उत्तर
Let y = `sqrt(((x - 1)(x - 2))/((x - 3)(x^2 + x + 1)))`
y = `(((x - 1)(x - 2))/((x - 3)(x^2 + x + 1)))^(1/2)`
Taking logarithm on both sides we get,
log y = `1/2` {[log(x – 1) + log(x – 2)] – [(log(x – 3) + log(x2 + x + 1)]}
log y = `1/2` [log(x – 1) + log(x – 2) – log(x – 3) – log(x2 + x + 1)]
Differentiating with respect to x,
`1/y "dy"/"dx" = 1/2[1/(x - 1) (1 - 0) + 1/(x - 2)(1 - 0) - 1/(x -3)(1 - 0) - 1/(x^2 + x + 1) (2x + 1)]`
`1/y "dy"/"dx" = 1/2[1/(x - 1) + 1/(x - 2) - 1/(x - 3) - (2x + 1)/(x^2 + x + 1)]`
`therefore "dy"/"dx" = 1/2 y [1/(x - 1) + 1/(x - 2) - 1/(x - 3) - (2x + 1)/(x^2 + x + 1)]`
`= 1/2 sqrt(((x - 1)(x - 2))/((x - 3)(x^2 + x + 1))) [1/(x - 1) + 1/(x - 2) - 1/(x - 3) - (2x + 1)/(x^2 + x + 1)]`
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