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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . Lecture 6 18.01 Fall 2006 Lecture 6: Exponential and Log, Logarithmic Differentiation, Hyperbolic Functions Taking the derivatives of exponentials and logarithms Background We always assume the base, a , is greater than 1. a = 1; a 1 = a ; a 2 = a a ; ... a x 1 + x 2 = a x 1 a x 2 ( a x 1 ) x 2 = a x 1 x 2 p q q a = a p (where p and q are integers) r To define a for real numbers r , fill in by continuity. d Todays main task: find a x dx We can write d a x + x x x a = lim a dx x x We can factor out the a x : x + x x x x lim a a = lim a x a 1 = a x lim a 1 x x x x x x Lets call M ( a ) lim a x 1 x x We dont yet know what M ( a ) is, but we can say d a x = M ( a ) a x dx Here are two ways to describe M ( a ): d 1. Analytically M ( a ) = a x at x = 0. dx Indeed, M ( a ) = lim a 0+ x a = d a x x x dx x =0 1 Lecture 6 18.01 Fall 2006 M(a) (slope of a x at x=0) a x Figure 1: Geometric definition of M ( a ) x 2. Geometrically, M ( a ) is the slope of the graph y = a at x = 0....
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lec6 - MIT OpenCourseWare http://ocw.mit.edu 18.01 Single...

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