5128_Ch03_pp098-184
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Section 3.9 Derivatives of Exponential and Logarithmic Functions 175<br />
Setting these two formulas for m equal to each other, we have<br />
ln a<br />
1 a a <br />
ln a 1<br />
e ln a e 1<br />
a e<br />
m 1 . Now try Exercise 31.<br />
e<br />
Derivative of log a x<br />
To find the derivative of log a x for an arbitrary base a 0, a 1, we use the changeof-base<br />
formula for logarithms to express log a x in terms of natural logarithms, as follows:<br />
ln<br />
x<br />
log a x .<br />
l n a<br />
The rest is easy:<br />
d d<br />
log d x a x d x ( ln<br />
ln<br />
x<br />
) a<br />
1 d<br />
• ln x<br />
ln a d x<br />
Since ln a is a constant<br />
1<br />
• 1 ln a x <br />
1<br />
.<br />
x l n a<br />
So, if u is a differentiable function of x and u 0, the formula is as follows.<br />
For a 0 and a 1,<br />
d<br />
1<br />
log d x a u d u<br />
. u l n a dx<br />
EXAMPLE 4 Going the Long Way with the Chain Rule<br />
Find dydx if y log a a sin x .<br />
SOLUTION<br />
Carefully working from the outside in, we apply the Chain Rule to get:<br />
d<br />
log d x a a sin x 1 d<br />
a<br />
sin x • a<br />
ln a d x<br />
sin x <br />
1<br />
a<br />
sin x • a<br />
ln a<br />
sin x d<br />
ln a • sin x d x<br />
log a u, u a sin x<br />
a u , u sinx<br />
a sin<br />
x<br />
ln<br />
a<br />
as<br />
in<br />
x • cos x<br />
ln<br />
a<br />
cos x. Now try Exercise 23.