derivative of ln(1-6x) — step-by-step solution

Let's perform the differentiation:

\frac{d}{d x} (\ln (1 - 6 x))

Apply the differentiation rule:

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

with f =

1 - 6 x

ⓘ The derivative of ln(x) is 1/x:

\frac{d}{d x} (\ln (1 - 6 x))

\frac{d}{d x} (1 - 6 x) \cdot \frac{1}{1 - 6 x}

\frac{d}{d x} (1 - 6 x) \cdot \frac{1}{1 - 6 x}

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{d}{d x} (1 - 6 x) \cdot \frac{1}{1 - 6 x}

\frac{\frac{d}{d x} (1 - 6 x) \cdot 1}{1 - 6 x}

\frac{\frac{d}{d x} (1 - 6 x)}{1 - 6 x}

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

\frac{d}{d x} (1 - 6 x)

0 + \frac{d}{d x} (- 6 x)

\frac{0 + \frac{d}{d x} (- 6 x)}{1 - 6 x}

ⓘ Adding 0 changes nothing:

0 + \frac{d}{d x} (- 6 x)

\frac{d}{d x} (- 6 x)

■A constant factor stays in front of the derivative, (k*u)' = k*u':

\frac{d}{d x} (- 6 x)

-6

- \frac{6}{1 - 6 x}

ⓘ The minus sign of the fraction goes into the numerator or the denominator, changing the sign of each of its terms:

- \frac{6}{1 - 6 x}

\frac{6}{- 1 + 6 x}

\frac{6}{- 1 + 6 x}

▶Derivative:

\frac{6}{6 x - 1}