derivative of ln(1/(1-x)) — step-by-step solution

To ease differentiation, the expression

\frac{1}{1 - x}

is turned into

{(1 - x)}^{-1}

Let's perform the differentiation:

\frac{d}{d x} (\ln ({(1 - x)}^{-1}))

Apply the differentiation rule:

\frac{d}{d x} (\ln (f)) = \frac{d}{d x} (f) \cdot \frac{1}{f}

with f =

{(1 - x)}^{-1}

ⓘ The derivative of ln(x) is 1/x:

\frac{d}{d x} (\ln ({(1 - x)}^{-1}))

\frac{d}{d x} ({(1 - x)}^{-1}) \cdot \frac{1}{{(1 - x)}^{-1}}

\frac{d}{d x} ({(1 - x)}^{-1}) \cdot \frac{1}{{(1 - x)}^{-1}}

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{d}{d x} ({(1 - x)}^{-1}) \cdot \frac{1}{{(1 - x)}^{-1}}

\frac{\frac{d}{d x} ({(1 - x)}^{-1}) \cdot 1}{{(1 - x)}^{-1}}

\frac{\frac{d}{d x} ({(1 - x)}^{-1})}{{(1 - x)}^{-1}}

■The factor is moved to the numerator or the denominator by changing the sign of its exponent, for a nonzero base:

\frac{\frac{d}{d x} ({(1 - x)}^{-1})}{{(1 - x)}^{-1}}

\frac{d}{d x} ({(1 - x)}^{-1}) \cdot {(1 - x)}^{1}

\frac{d}{d x} ({(1 - x)}^{-1}) \cdot {(1 - x)}^{1}

Apply the differentiation rule:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{n - 1}

with f =

1 - x

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d x} ({(1 - x)}^{-1})

-1 \cdot \frac{d}{d x} (1 - x) \cdot {(1 - x)}^{-2}

-1 \cdot \frac{d}{d x} (1 - x) \cdot {(1 - x)}^{-2} \cdot (1 - x)

ⓘ Multiplying by -1 gives the opposite:

-1 \cdot \frac{d}{d x} (1 - x) \cdot {(1 - x)}^{-2} \cdot (1 - x)

- \frac{d}{d x} (1 - x) \cdot {(1 - x)}^{-2} \cdot (1 - x)

- \frac{d}{d x} (1 - x) \cdot {(1 - x)}^{-2} \cdot (1 - x)

ⓘ The factor is moved to the numerator or the denominator by changing the sign of its exponent, for a nonzero base:

- \frac{d}{d x} (1 - x) \cdot {(1 - x)}^{-2} \cdot (1 - x)

- \frac{\frac{d}{d x} (1 - x) \cdot (1 - x)}{{(1 - x)}^{2}}

- \frac{\frac{d}{d x} (1 - x) \cdot (1 - x)}{{(1 - x)}^{2}}

The fraction:

\frac{1 - x}{{(1 - x)}^{2}}

simplifies by the common factor:

1 - x

into the new quotient:

\frac{1}{1 - x}

- \frac{\frac{d}{d x} (1 - x)}{1 - x}

ⓘ The derivative of a sum is the sum of the derivatives:

\frac{d}{d x} (1 - x)

0 + \frac{d}{d x} (- x)

- \frac{0 + \frac{d}{d x} (- x)}{1 - x}

ⓘ Adding 0 changes nothing:

0 + \frac{d}{d x} (- x)

\frac{d}{d x} (- x)

■The derivative of x is 1:

\frac{d}{d x} (- x)

-1

- (\frac{-1}{1 - x})

ⓘ Two minus signs cancel out:

- (\frac{-1}{1 - x})

\frac{1}{1 - x}

▶Derivative:

\frac{1}{1 - x}