Example Function:\[f(x)={\sec{\left( {\tan {x} }+{\left( {\sec {x} } \right)}{x}^{{2}}\right)}}\]
Derivative:\[f'(x)={\left( {\sec{\left( {\tan {x} }+{\left( {\sec {x} } \right)}{x}^{{2}}\right)}} \right)}{\tan{\left( {\tan {x} }+{\left( {\sec {x} } \right)}{x}^{{2}}\right)}}{\left({\left( {\sec {x} } \right)}^{{2}}+{\left( {\sec {x} } \right)}{\tan {x} }{x}^{{2}}+{2}{x}{\sec {x} }\right)}\]
Hints about how the derivative was found:
\({\frac{d}{dx}\left[{\sec{\left( {\tan {x} }+{\left( {\sec {x} } \right)}{x}^{{2}}\right)}}\right]}\)=\({\left( {\sec{\left( {\tan {x} }+{\left( {\sec {x} } \right)}{x}^{{2}}\right)}} \right)}{\tan{\left( {\tan {x} }+{\left( {\sec {x} } \right)}{x}^{{2}}\right)}}{\frac{d}{dx}\left[{\tan {x} }+{\left( {\sec {x} } \right)}{x}^{{2}}\right]}\)

\({\frac{d}{dx}\left[{x}^{{2}}\right]}\)=\({2}{x}\)

\({\frac{d}{dx}\left[{\left( {\sec {x} } \right)}{x}^{{2}}\right]}\)=\({x}^{{2}}{\frac{d}{dx}\left[{\sec {x} }\right]}+{\left( {\sec {x} } \right)}{\frac{d}{dx}\left[{x}^{{2}}\right]}\)





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