Calculus of a Single Variable, Chapter 5, 5.1, Section 5.1, Problem 64

y=ln|secx+tanx|
To take the derivative of this function, use the formula:
(ln u)' = 1/u* u'
Applying this formula, y' will be:
y' = 1/(secx+tanx) * (secx+tanx)'
To get the derivative of the inner function, use the formulas:
(sec theta)'= sec theta tan theta
(tan theta)' =sec^2 theta
So y' will become:
y' = 1/(secx +tanx) * (secxtanx+sec^2x)
Simplifying it, the derivative will be:
y'=(secxtanx+sec^2x)/(secx+tanx)
y'=(secx(tanx+secx))/(secx + tanx)
y'=(secx(secx+tanx))/(secx+tanx)
y'=secx
Therefore, the derivative of the given function is y' =secx .