Question

Find the n^th order derivative of log x.

Answer 1

Let y = log x

Differentiate w. r. t. x

`dy/dx = d/dx(log x) = 1/x`

Differentiate w. r. t. x

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

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

Differentiate w. r. t. x

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

`(d^3y)/(dx^3) = (-1)^1((-2)/x^3) = ((-1)^2 * 1 * 2)/x^3`

In general, the n^th order derivative will be

`(d^ny)/(dx^n) = ((-1)^(n - 1)*1*2*3... (n - 1))/x^n`

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