Solve d/dx ( ln (x^(-1)))

Evaluate

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

Use the chain rule \frac{d}{d x} (f (g)) = \frac{d}{d g} (f (g)) \times \frac{d}{d x} (g) where the g = x^{- 1}, to find the derivative

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

Use \frac{d}{d x} ln x = \frac{1}{x} to find derivative

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

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

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

Substitute back

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

Calculate

More Steps

x (- x^{- 2})

Express with a positive exponent using a^{- n} = \frac{1}{a ^{n}}

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

Multiplying or dividing an odd number of negative terms equals a negative

- x \times \frac{1}{x ^{2}}

Cancel out the common factor x

- 1 \times \frac{1}{x}

Solution

- \frac{1}{x}

Evaluate the derivative