Derivatives of General Exponential Functions
(y=bx)
Recall the derivative of 𝑓(𝑥) = 𝑏 𝑥 from first principles:
𝑓
If b= 2, ∴
𝑑
𝑑𝑥
′ (𝑥)
𝑏ℎ − 1
= 𝑏 (lim
)
ℎ→0
ℎ
When b=e, this
limit equals 1!
𝑥
(2𝑥 ) ≐ (0.69)2𝑥 and if b = 3, ∴
Calculate:
𝑙𝑛2 =
𝑑
𝑑𝑥
(3𝑥 ) ≐ (1.1)3𝑥 .
What do you notice??????
𝑙𝑛3 =
Therefore, the derivative function, 𝑓 ′ (𝑥), of 𝑓(𝑥) = 𝑏 𝑥 becomes:
𝑓
′ (𝑥)
𝑏ℎ − 1
= 𝑏 (lim
)
ℎ→0
ℎ
𝑥
=
DERIVATIVE of f(x) = bx
For the function f x   b x , f ' x   b x  ln b .
If f x   b g x  , then f ' x   b g x   g' x   ln b .
Example 
Determine the derivative of each of the following:
a)
y  5x
c)
y  3x
2 2 x
2
b)
y  3x
d)
y  x3 3x
 
Example 
Determine the equation of the tangent to f x   2 3 x at x = 2.
Example 
Determine the critical points of f ( x )  10 2 x e x .
Example 
A certain radioactive substance decays exponentially over time. The
amount of a sample of the substance that remains, P, after t years is
given by 𝑃(𝑡) = 100(1.23)−5𝑡 , where P is expressed as a percent.
2
a)
Determine the rate of change of the function, 𝑃 ′ (𝑡).
b)
Determine the rate of decay when 50% of the original sample has
decayed.
Homework: p.240 #1–6, 8