"We will do this by the Chain Rule."
"Recall:" \qquad \qquad \qquad y \ = \ ln | x | \qquad rArr \qquad y' \ = \ 1/x.
"So, by the Chain Rule:"
\qquad \quad y \ = \ ln | f(x) | \qquad rArr \qquad y' \ = \ [ 1/f(x) ] cdot f'(x) \ = \ { f'(x) }/f(x).
\qquad :. \qquad \qquad y \ = \ ln | f(x) | \qquad rArr \qquad y' \ = \ { f'(x) }/f(x).
"So, in our example:"
\qquad \qquad \qquad \qquad \qquad y \ = \ ln| secx + tanx |; \qquad \qquad \quad \ \color{blue}{ f(x) \ = \ secx + tanx }
\qquad \qquad \qquad \qquad \quad y' \ = \ { ( secx + tanx )' }/{ secx + tanx }; \qquad \qquad \quad \color{blue}{ = { \ f'(x) }/f(x) }
\qquad \qquad \qquad \qquad \qquad \quad \ = \ { ( secx )'+ ( tanx )' }/{ secx + tanx };
\qquad \qquad \qquad \qquad \qquad \quad \ = \ { secx tanx+ sec^2x }/{ secx + tanx };
\qquad \qquad \qquad \qquad \qquad \quad \ = \ { secx ( tanx+ secx ) }/{ secx + tanx };
\qquad \qquad \qquad \qquad \qquad \quad \ = \ { secx ( secx + tanx ) }/{ secx + tanx };
\qquad \qquad \qquad \qquad \qquad \quad \ = \ { secx color{red}cancel{ ( secx + tanx ) } }/color{red}cancel{ ( secx + tanx ) };
\qquad \qquad \qquad \qquad \qquad \quad \ = \ secx.
"So, we have now:"
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad y' \ = \ secx.
"Summarizing:"
\qquad \qquad \qquad \qquad y \ = \ ln| secx + tanx | \qquad rArr \qquad y' \ = \ secx.
