On-line Math 21

3.3.4  Derivatives of other Exponential and Logarithmic functions

  Derivatives of exponential functions

Example 2 If

f(x) = (x2+3)4e3x (x-3)(2x+5)2 ,

find f¢(x) .

Solution

ln(f(x)) = ln æ ç è (x2+3)4e3x (x-3)(2x+5)2 ö ÷ ø = ln( (x2+3)4e3x) -ln( (x-3)(2x+5)2) = 4ln( x2+3) +3xln( e) -ln( x-3) -2ln( 2x+5) = 4ln( x2+3) +3x-ln( x-3) -2ln( 2x+5) ,

so

d dx ln(f(x)) = ( 4ln( x2+3) +3x-ln( x-3) -2ln( 2x+5) ) ¢ = 8x x2+3 +3- 1 x-3 - 4 2x+5 ,

which means that

df dx = f(x) æ ç è d dx ln(f(x)) ö ÷ ø = æ ç è (x2+3)4e3x (x-3)(2x+5)2 ö ÷ ø æ ç è 8x x2+3 +3- 1 x-3 - 4 2x+5 ö ÷ ø .

This may not be terribly pretty, but it is simpler than doing it directly.

Copyright (c) 2000 by David L. Johnson.