What is the derivative of ln(cos(ln(x))) ?

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

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

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

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العربية

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

- \frac{tan ( ln ( x ) )}{x}

Solution steps

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

Apply the chain rule:

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

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

\frac{d}{d x} (cos (ln (x))) = - \frac{sin ( ln ( x ) )}{x}

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

Simplify

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

= - \frac{tan ( ln ( x ) )}{x}

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