Calculus Lecture Notes Section 5.4, 5.5 Page 1 of 7
Choi Ver. 1
Section 5.4 Differentiation of Exponential Functions
Example 1: Find the derivative of each of the following functions:
𝑎)##𝑓
(
𝑥
)
= 𝑥(𝑒*###########################𝑏)##𝑓
(
𝑥
)
=
(
𝑒*+ 5
)
.
Example 2: Find the derivative of each of the following functions:
𝑎)##𝑓
(
𝑥
)
= 𝑒/*############################𝑏)##𝑓
(
𝑥
)
= 𝑒01*###########################𝑐)##𝑓
(
𝑥
)
= 𝑒34567(5############################𝑑)##𝑓
(
𝑥
)
= 4𝑒:573
Example 3: Find the derivative of each of the following functions:
𝑎)##𝑓
(
𝑥
)
= 𝑥𝑒0(*###############################################𝑏)##𝑓
(
𝑥
)
=𝑒*
𝑒*+ 𝑒0* #############################################𝑐)#𝑓
(
𝑥
)
=
(
𝑥;+ 1
)
𝑒(*7;
Rule 1: Derivative of the Exponential Function
𝑑
𝑑𝑥 (𝑒*) = 𝑒*
Rule 2: Chain Rule for Exponential Functions
If f(x) is a differentiable function, then
𝑑
𝑑𝑥 (𝑒=(*)) = 𝑒=(*)𝑓′(𝑥)
Calculus Lecture Notes Section 5.4, 5.5 Page 2 of 7
Example 4: Determine the intervals of concavity for the function
𝑓
(
𝑥
)
= 𝑥𝑒*
.
Example 5: Find the absolute extrema of the function
𝑓
(
𝑥
)
= 𝑒*601##𝑜𝑛##[−2, 2]
.
Example 6: The price of a commodity in dollars per unit at time t (measured in weeks) is given by
