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

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

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

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

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

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

Solution steps

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

Apply the chain rule:

\frac{1}{\frac{s i n ( x )}{x}} \frac{d}{d x} (\frac{sin ( x )}{x})

= \frac{1}{\frac{s i n ( x )}{x}} \frac{d}{d x} (\frac{sin ( x )}{x})

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

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

Simplify

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

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

y = 4 ln (x), y = x ln (x)

x \to 4 lim (e^{\frac{3}{4 - x}})

\frac{\partial}{\partial x} (x^{2} - 3 x y + y^{2})

\int 3 sin (ln (x)) d x

\int_{0}^{x} e^{t} d t

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