Question

Differentiate the following w.r.t. x :

`(x +  1)^2/((x + 2)^3(x + 3)^4`

Answer 1

Let y = `(x + 1)^2/((x + 2)^3(x + 3)^4`

Then, log y = `log[(x + 1)^2/((x + 2)^3(x + 3)^4)]`

= log(x + 1)² – log(x + 2)³ – log(x + 3)⁴

= 2log(x + 1) – 3log(x + 2) – 4log(x + 3)

Differentiating w.r.t. x, we get

`(1)/y "dy"/"dx" = 2"d"/"dx"[log(x + 1)] -3"d"/"dx"[log(x + 2)] - 4"d"/"dx"[log(x + 3)]`

= `2 xx (1)/(x + 1)."d"/"dx"(x + 1) -3 xx (1)/(x + 2)."d"/"dx"(x + 2) - 4 xx (1)/(x + 3)."d"/"dx"(x + 3)`

= `(2)/(x + 1).(1 + 0) - (3)/(x + 2).(1 + 0) - (4)/(x + 3).(1 + 0)`

∴ `"dy"/"dx" = y[2/(x + 1) - 3/(x + 2) - 4/(x + 3)]`

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