Question

Differentiate the following with respect to x.

`sqrt(((x - 1)(x - 2))/((x - 3)(x^2 + x + 1)))`

Answer 1

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)]`