Differentiate the following natural logarithmic functions, using the rule found in (11.6):

Note how a multiplicative constant within the log expression drops out in differentiation as in multiplicative constant outside the log expression as in

(f) remains.

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d/dx[lng(x)] = g'(x). g(x) (a) y = In 7x3 Let g(x)=7x3, then g'(x) 21x2 and from (11.6), y' = 7+13 (21x) = 3x 7.x3 (c) y In(8x+3) = 3 x y': = 1 8x2 +3 (16x) 16.x (b) yIn(4x+9) Let g(x)=4x+9, then g'(x) 4 and from (11.6), = y' = = 1 4 (4) = 4.x +9 4.x +9 (d) y In(x + 7x+15) 1 y' = (2x+7) x+7x+15 = 2x+7 x+7x+15 = 8.x +3 (e) y In 18.x = 1 18.x (18) = (f) y = 18lnx y' = 18.(1)= x 18 x