What is the derivative of ln(tan(x/2+pi/4)) ?

The derivative of ln(tan(x/2+pi/4)) is sec(x)

What is the first derivative of ln(tan(x/2+pi/4)) ?

The first derivative of ln(tan(x/2+pi/4)) is sec(x)

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\frac{d}{d x} (ln (tan (\frac{x}{2} + \frac{π}{4})))

sec (x)

Solution steps

\frac{d}{d x} (ln (tan (\frac{x}{2} + \frac{π}{4})))

Simplify

ln (tan (\frac{x}{2} + \frac{π}{4})) : ln (\frac{sin ( \frac{x}{2} ) + cos ( \frac{x}{2} )}{cos ( \frac{x}{2} ) - sin ( \frac{x}{2} )})

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

Apply the chain rule:

\frac{1}{\frac{s i n ( \frac{x}{2} ) + c o s ( \frac{x}{2} )}{c o s ( \frac{x}{2} ) - s i n ( \frac{x}{2} )}} \frac{d}{d x} (\frac{sin ( \frac{x}{2} ) + cos ( \frac{x}{2} )}{cos ( \frac{x}{2} ) - sin ( \frac{x}{2} )})

= \frac{1}{\frac{s i n ( \frac{x}{2} ) + c o s ( \frac{x}{2} )}{c o s ( \frac{x}{2} ) - s i n ( \frac{x}{2} )}} \frac{d}{d x} (\frac{sin ( \frac{x}{2} ) + cos ( \frac{x}{2} )}{cos ( \frac{x}{2} ) - sin ( \frac{x}{2} )})

\frac{d}{d x} (\frac{sin ( \frac{x}{2} ) + cos ( \frac{x}{2} )}{cos ( \frac{x}{2} ) - sin ( \frac{x}{2} )}) = \frac{1}{( cos ( \frac{x}{2} ) - sin ( \frac{x}{2} ) ) ^{2}}

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

Simplify

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

= sec (x)

What is the derivative of ln(tan(x/2+pi/4)) ?
The derivative of ln(tan(x/2+pi/4)) is sec(x)
What is the first derivative of ln(tan(x/2+pi/4)) ?
The first derivative of ln(tan(x/2+pi/4)) is sec(x)