Trigonometry and Rules of Differentiation lecture notes - MA103 Week 5 Trigonometry and Rules of Dierentiation 1 Rules of Dierentiation(Text 3.1 3.2

Trigonometry and Rules of Differentiation lecture notes -...

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MA103 Week 5 Trigonometry and Rules of Differentiation 1. Rules of Differentiation (Text: 3.1, 3.2) Sum/Difference Rule: [ f ( x ) ± g ( x )] 0 = f 0 ( x ) ± g 0 ( x ) Product Rule: [ f ( x ) · g ( x )] 0 = f 0 ( x ) · g ( x ) + f ( x ) · g 0 ( x ) Constant Multiple Rule: [ c · f ( x )] 0 = c · f 0 ( x ), c R Power Rule: ( x r ) 0 = rx r - 1 , r R and x r - 1 defined Quotient Rule: f ( x ) g ( x ) 0 = f 0 ( x ) g ( x ) - f ( x ) g 0 ( x ) [ g ( x )] 2 2. Derivatives of Exponential Functions (Text: 3.1) From the limit definition, the derivative of an exponential function f ( x ) = a x is given by f 0 ( x ) = lim h 0 a x + h - a x h = a x lim h 0 a h - 1 h . We define e to be the number for which lim h 0 e h - 1 h = 1, and so d dx ( e x ) = e x lim h 0 e h - 1 h = e x . Applying the chain rule, we have d dx e f ( x ) = e f ( x ) · f 0 ( x ). Looking at the general exponential function y = a x : d dx [ a x ] = d dx e (ln a ) x = e (ln a ) x · ln a = a x · ln a and the Chain Rule gives d dx a f ( x ) = a f ( x ) · ln a · f 0 ( x ). 3. Trigonometry (Appendix D) The inside front cover of the text contains a summary of important properties and identities involving trigono- metric functions. You should become familiar with the material provided there.

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