If a > 0, a ≠ 1, f(x) = ax is a continuous function with domain R and range (0, ∞).
In particular, ax > 0 for all x.
If 0 < a < 1, f(x) = ax is a decreasing function.
If a > 1, f is an increasing function.
There are basically three kinds of exponential functions y = ax.
Since (1/a)x = 1/ax = a-x, the graph of y = (1/a)x is just
the reflection of the graph of y = ax about the y-axis.
The exponential function occurs very frequently in mathematical
models of nature and society.
Derivative of exponential function 𝑓 𝑥 = 𝑒 𝑥
𝑓′
𝑓 𝑥 + ℎ − 𝑓(𝑥)
𝑒 𝑥+ℎ − 𝑒 𝑥
𝑒 𝑥𝑒ℎ − 𝑒 𝑥
𝑥 = lim
= lim
= lim
ℎ→0
ℎ→0
ℎ→0
𝑥
ℎ
ℎ
ℎ
𝑒 𝑥 (𝑒 ℎ −1)
𝑒
−1
= lim
= 𝑒 𝑥 lim
= 𝑒 𝑥 𝑓′(0)
ℎ→0
ℎ→0
ℎ
ℎ
By definition, this is derivative 𝑓′(0) ,
what is the slope of 𝑒 𝑥 at 𝑥 = 1.
1
𝑒ℎ − 1
ℎ
1.71828
0.1
1.05171
0.01
1.00502
0.001
1.00050
0.0001
1.00005
0.00001
1.00001
ℎ
𝑒ℎ − 1
lim
=1
ℎ→0
ℎ
𝑑 𝑥
𝑒 = 𝑒𝑥
𝑑𝑥
example:
Differentiate the function y = e tan x
To use the Chain Rule, we let u = tan x.
Then, we have y = eu.
example:
Find y’ if y = e-4x sin 5x.
chain rule:
d u
u du
e e
dx
dx
We can now use this formula to find the derivative of
𝑦 = 𝑎𝑥
𝑑 𝑥
𝑎
𝑑𝑥
=
𝑑 𝑥
𝑎
𝑑𝑥
⟹
𝑑
𝑑𝑥
ln 𝑦 = ln 𝑎 𝑥 = 𝑥 ln 𝑎
(𝑒 𝑥 ln 𝑎 ) = 𝑒 𝑥 ln 𝑎
= 𝑎 𝑥 ∙ ln 𝑎
𝑑
𝑑𝑥
a
⟹ 𝑎 𝑥 = 𝑒 𝑥 ln 𝑎
(𝑥 ln 𝑎) = 𝑒 𝑥 ln 𝑎 ∙ ln 𝑎
x
𝑦 = ln 𝑥
𝑒𝑦 = 𝑥 ⟹
𝑑 𝑦
𝑑
𝑒 =
𝑥
𝑑𝑥
𝑑𝑥
𝑒𝑦
𝑑𝑦
𝑑𝑥
𝑑
1
ln 𝑥 =
𝑑𝑥
𝑥
= 1
⟹
⟹𝑥
𝑑𝑦
𝑑𝑥
= 1 ⟹
𝑑𝑦
𝑑𝑥
=
1
𝑥
example:
Differentiate y = ln(x3 + 1).
To use the Chain Rule, we let u = x3 + 1.
Then, y = ln u.
example:
Find:
example:
Differentiate f ( x ) 
ln x
example:
If we first simplify the given function using the laws of logarithms, the
differentiation becomes easier
example:
Find f ’(x) if 𝑓(𝑥) = ln |𝑥|.
Thus, f ’(x) = 1/x for all x ≠ 0.
The result is worth remembering:
a logarithmic function with base a in terms
of the natural logarithmic function:
Since ln a is a constant, we can differentiate as follows:
example:
IMPORTANT and UNUSUAL: If you have a daunting task to find derivative
in the case of a function raised to the function 𝑥 𝑥 , 𝑥 sin 𝑥 … , , or a crazy
product, quotient, chain problem you do a simple trick:
FIRST find logarithm , 𝑙𝑛, so you’ll have sum instead of product, and
product instead of exponent. Life will be much, much easier.
STEPS IN LOGARITHMIC DIFFERENTIATION
1. Take natural logarithms of both sides of an equation y = f(x) and
use the Laws of Logarithms to simplify.
2. Differentiate implicitly with respect to x.
3. Solve the resulting equation for y’.
example:
Differentiate:
1.
2.
3.
Since we have an explicit expression for y, we can substitute and write
If we hadn’t used logarithmic differentiation the resulting calculation
would have been horrendous.
example:
𝑦 = 𝑥 sin 𝑥
𝑦 ′ =?
1 ′
sin 𝑥
ln 𝑦 = (sin 𝑥) ln 𝑥 ⇒
𝑦 = (cos 𝑥) ln 𝑥 +
𝑦
𝑥
𝑦 ′ = (ln 𝑥)𝑥 sin 𝑥 cos 𝑥 + (sin 𝑥) 𝑥 sin 𝑥−1
Try: 𝑦 = (sin 𝑥) 𝑥