What is the derivative of ln(cos((x-1/x))) ?

The derivative of ln(cos((x-1/x))) is -(tan((x-1)/x))/(x^2)

What is the first derivative of ln(cos((x-1/x))) ?

The first derivative of ln(cos((x-1/x))) is -(tan((x-1)/x))/(x^2)

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

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

Solution steps

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

Apply the chain rule:

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

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

\frac{d}{d x} (cos (\frac{x - 1}{x})) = - \frac{sin ( \frac{x - 1}{x} )}{x ^{2}}

= \frac{1}{cos ( \frac{x - 1}{x} )} (- \frac{sin ( \frac{x - 1}{x} )}{x ^{2}})

Simplify

\frac{1}{cos ( \frac{x - 1}{x} )} (- \frac{sin ( \frac{x - 1}{x} )}{x ^{2}}) : - \frac{tan ( \frac{x - 1}{x} )}{x ^{2}}

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

What is the derivative of ln(cos((x-1/x))) ?
The derivative of ln(cos((x-1/x))) is -(tan((x-1)/x))/(x^2)
What is the first derivative of ln(cos((x-1/x))) ?
The first derivative of ln(cos((x-1/x))) is -(tan((x-1)/x))/(x^2)