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

The derivative of ln((cos(x)/x)) is (-xsin(x)-cos(x))/(xcos(x))

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

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

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

\frac{- x sin ( x ) - cos ( x )}{x cos ( x )}

Solution steps

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

Apply the chain rule:

\frac{1}{\frac{c o s ( x )}{x}} \frac{d}{d x} (\frac{cos ( x )}{x})

= \frac{1}{\frac{c o s ( x )}{x}} \frac{d}{d x} (\frac{cos ( x )}{x})

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

= \frac{1}{\frac{c o s ( x )}{x}} \cdot \frac{- x sin ( x ) - cos ( x )}{x ^{2}}

Simplify

\frac{1}{\frac{c o s ( x )}{x}} \cdot \frac{- x sin ( x ) - cos ( x )}{x ^{2}} : \frac{- x sin ( x ) - cos ( x )}{x cos ( x )}

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

\frac{d}{d x} ((4 x + 5 x^{2})^{4})

\int \frac{16 - x ^{2}}{x ^{2}} d x

\int \frac{1}{8 x - x ^{2}} d x

5 3 x

\frac{\partial}{\partial y} (x ln (yz))

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