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

The derivative of (ln(x)/(sin(x))) is (csc(x))/x-cot(x)csc(x)ln(x)

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

The first derivative of (ln(x)/(sin(x))) is (csc(x))/x-cot(x)csc(x)ln(x)

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

\frac{csc ( x )}{x} - cot (x) csc (x) ln (x)

Solution steps

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

Simplify

\frac{ln ( x )}{sin ( x )} : ln (x) csc (x)

= \frac{d}{d x} (ln (x) csc (x))

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = ln (x), g = csc (x)

= \frac{d}{d x} (ln (x)) csc (x) + \frac{d}{d x} (csc (x)) ln (x)

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

\frac{d}{d x} (csc (x)) = - cot (x) csc (x)

= \frac{1}{x} csc (x) + (- cot (x) csc (x)) ln (x)

Simplify

\frac{1}{x} csc (x) + (- cot (x) csc (x)) ln (x) : \frac{csc ( x )}{x} - cot (x) csc (x) ln (x)

= \frac{csc ( x )}{x} - cot (x) csc (x) ln (x)

\int \frac{1}{( 2 - 3 x )} d x

\int \frac{7}{x ^{2}} d x

\int 20 x^{3} d x

\frac{d Q}{d t} = 0.03 (1 - Q)^{\frac{2}{3}}

n = 1 \sum \infty \frac{e ^{nπ}}{π ^{n e}}

What is the derivative of (ln(x)/(sin(x))) ?
The derivative of (ln(x)/(sin(x))) is (csc(x))/x-cot(x)csc(x)ln(x)
What is the first derivative of (ln(x)/(sin(x))) ?
The first derivative of (ln(x)/(sin(x))) is (csc(x))/x-cot(x)csc(x)ln(x)