3.3. DERIVATIVES OF THE LIBRARY OF FUNCTIONS 95
Knowledge Box 3.5
e general power rule for derivatives
If f .x/ D x
r
for any real number r, then
f
0
.x/ D rx
r1
Example 3.39 Find the derivative of f .x/ D
p
x.
Solution:
f .x/ D
p
x D x
1=2
Apply the power rule and we get
f
0
.x/ D
1
2
x
1=21
D
1
2
x
1=2
D
1
2x
1=2
D
1
2
p
x
˙
At this point we just take off and give a whole bunch of derivative rules. It’s hard to do good
examples until we get to Section 3.4 where we get the rules for combining functions in various
ways.
3.3.1 LOGS AND EXPONENTS
e rules for logarithm functions provide a sense of why ln.x/ is called the natural logarithm.
All the other logarithm functions have more complex derivative rules based on ln.x/.
Knowledge Box 3.6
Derivatives of log functions
• If f .x/ D ln.x/, then f
0
.x/ D
1
x
.
• If f .x/ D log
b
.x/, then f
0
.x/ D
1
x ln.b/
.
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