96 3. LIMITS, DERIVATIVES, RULES, AND THE MEANING OF THE DERIVATIVE
e exponential function y D e
x
has the simplest imaginable derivative.
Knowledge Box 3.7
Derivatives of exponential functions
• If f .x/ D e
x
, then f
0
.x/ D e
x
.
• If f .x/ D a
x
, then f
0
.x/ D ln.a/ a
x
.
Example 3.40 Find the derivative of f .x/ D 3
x
:
Solution:
f
0
.x/ D ln.3/ 3
x
˙
3.3.2 THE TRIGONOMETRIC FUNCTIONS
e derivatives of the trigonometric functions should be committed to memory. Some patterns
that help with this are as follows.
• Only the derivatives of co-functions are negative.
• A function and its co-function have derivatives that are obtained by replacing functions
with co-functions.
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