Calculus 220 section 4 5 The Derivative of ln x notes by Tim Pilachowski The natural logarithm function y ln x is the inverse of the natural exponential function y e x In Lecture 4 2 we determined that the slope d x of y e x is 1 at the point 0 1 i e e 1 By symmetry the x 0 dx slope of y ln x should also be 1 at the point 1 0 i e d ln x 1 Also recall that y ln x is increasing and concave x 1 dx 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 e ln x x then d ln x d e x 1 By the chain rule d e ln x e ln x d ln x x d ln x 1 d ln x 1 dx dx dx dx dx dx x 1 d d2 d 1 Note that ln x ln x 0 and 2 ln x 12 0 for all x in the domain 1 Also x 1 x x 1 dx x dx dx x 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 h x x 3 ln x find the first and second derivative Answers x 2 1 3 ln x x 5 6 ln x x3 x 2 3 ln x 1 Example B Given f x find the first derivative Answer ln x ln x 2 Example C Given g x ln x x 3 find the first derivative Answer 1 3 ln x x4 Carefully note the placement of coefficients when finding derivatives constant multiple rule chain rule m x k ln x n x ln k x Example D Give h x ln x 3 find the first and second derivatives Answers 3 3 x x2 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 h x ln x 5 find the first and second derivatives Answers 5 ln x 4 20 ln x 3 5 ln x 4 x x2 We can use the above to sketch the graph of h x ln x 5 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 e 4 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 4990
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