Question

Find the n^th derivative of the following:

`(1)/x`

Answer 1

Let y = `(1)/x`

Then `"dy"/"dx" = "d"/"dx"(1/x)`

= `-(1)/x^2`

= `((-1)^1 1!)/x^2`

`(d^2y)/(dx^2) = "d"/"dx"(-1/x^2)`

= `1"d"/"dx"(x^-2)`

= ( – 1)(– 2)x^–3 

= `((-1)^2. 1.2)/x^3`

= `((-1)^2 2!)/x^3`

`(d^3y)/(dx^3) = "d"/"dx"[((-1)^2. 2!)/x^3]`

= `(-1)^2.  2!"d"/"dx"(x^-3)`

= ( –1)². 2!.( – 3)x^–4

= `((-1)^3 xx 3.2!)/x^4`

= `((-1)^3. 3!)/x^4`

In general, the n^th order derivative is given by

`(d^ny)/(dx^n) = ((-1)^n.  n!)/(x^(n + 1)`.