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

The derivative of ln((sin(x)/(cos(x)))) is 2csc(2x)

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

The first derivative of ln((sin(x)/(cos(x)))) is 2csc(2x)

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

2 csc (2 x)

Solution steps

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

Simplify

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

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

Apply the chain rule:

\frac{1}{tan ( x )} \frac{d}{d x} (tan (x))

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

\frac{d}{d x} (tan (x)) = sec^{2} (x)

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

Simplify

\frac{1}{tan ( x )} sec^{2} (x) : 2 csc (2 x)

= 2 csc (2 x)

What is the derivative of ln((sin(x)/(cos(x)))) ?
The derivative of ln((sin(x)/(cos(x)))) is 2csc(2x)
What is the first derivative of ln((sin(x)/(cos(x)))) ?
The first derivative of ln((sin(x)/(cos(x)))) is 2csc(2x)