What is the first second and third derivative of ln 1-x?

Answer 1

Original function: f(x) = ln(1-x)

Using the chain rule for logarithms: (ln(u))' = 1/u * u'

The first derivative: f'(x) = 1/(1-x) * -1

= 1/(-1*(1-x))

= 1/(x-1)

= (x-1)^(-1)

Using the chain rule for polynomials: (au^n)' = nau^(n-1) * u'

The second derivative: f''(x) = -1*(x-1)^-2 * 1

= -(x-1)^-2

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

Using the chain rule again:

The third derivative: f'''(x) = 2*(x-1)^-3 * 1

= 2(x-1)^-3

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