



These
are a bit more involved than exponents with e as a base,



but not
THAT much. 





The
derivative of 2 ^{X} is 2
^{X }ln X. 





The
general rule is: 








Here's
why, start with: 


Y
= a^{X}



Take the
log (ln that is) of both sides: 


ln
Y = ln a^{X}



Use the
log with exponents trick: 


ln Y = X ln a 


Now solve
for X: 








ln a is a
constant. ln Y is a variable. 


So it's
easier to see how this will work, we will separate the two. 











Now we
take the derivative of X with respect to Y. 








WHAT?
Where did that 1/Y come from? 


Don't you
remember the last page? 


The
derivative of ln Y is 1/Y. 








Our
original problem was Y = aX, so substitute that in for Y ... 











But we
wanted dY/dX not dX/dY! 





OK, so
flip it over. 








The
original equation was Y = a^{X}. That means that if: 








And there
it is! 





copyright 2005 Bruce Kirkpatrick 
