Calculus 220, section 4.5 The Derivative of ln(x) notes by Tim Pilachowski The natural logarithm function, ()xy ln=, is the inverse of the natural exponential function, xey =. In Lecture 4.2 we determined that the slope of xey = is 1 at the point (0, 1), i.e. ()10==xedxdx. By symmetry, the slope of ()xy ln= should also be 1 at the point (1, 0), i.e. ( )11ln ==xxdxd. Also recall that ()xy ln= is increasing and concave down over its entire domain The formula we use for the derivative of ln(x) must meet all of these conditions. Finding a derivative formula for ln(x) is actually quite simple. First note that since xex=ln, then ()( )1ln== xdxdedxdx. By the chain rule, ()( ) ( ) ( )xxdxdxdxdxxdxdeedxdxx1ln1lnlnlnln=→=∗=∗= . Note that ( )1111ln ==== xxxxdxd. Also ( )01ln >=xxdxd and ( )01ln222<−=xxdxd for all x in the domain of ln(x). In other words, all of the necessary conditions listed above have been met. The examples below will utilize this formula along with the product rule, quotient rule and chain rule. Example A: Given ()xxxh ln3∗= find the first and second derivative. Answers: ( )xx ln312+; ()xx ln65 + Example B: Given ( )xxxfln3= find the first derivative. Answer: ()[ ]22ln1ln3xxx − Example C: Given ( )3lnxxxg = find the first derivative. Answer: 4ln31xx−Carefully note the placement of coefficients when finding derivatives. constant multiple rule chain rule ()()xkxm ln= ()()xkxn ln= Example D: Give ()()3ln xxh =, find the first and second derivatives. Answers: 23;3xx− When using the chain rule, it is extremely important to correctly identify the “outside” and “inside” functions. Check that your composition is set up correctly. Example E: Give ()[]5ln xxh =, find the first and second derivatives. Answers: ()xx4ln5; ()()243ln5ln20xxx −We can use the above to sketch the graph of ()[]5ln xxh =. asymptotes: y-intercept: x-intercepts: possible extrema: possible points of inflection: A table of signs tells us concavity: interval sign of h′′ concavity The point of inflection at (4e, 1024) ≈ (54.598, 1024) is far outside the standard window on a graphing calculator. One could zoom out a good bit, or use different scales for x- and y-axes—but then the other characteristics of the graph would be obscured. Note the three versions of the graph pictured below. window [–1, 19] by [–10, 10] window [–1, 1999] by [–10, 1990] window [–1, 499] by [–10,
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