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 𝑓𝑓 𝑥𝑥 = 𝑒𝑒 𝑥𝑥
𝑓𝑓 ′
1
𝑓𝑓 𝑥𝑥 + ℎ − 𝑓𝑓(𝑥𝑥)
𝑒𝑒 𝑥𝑥+ℎ − 𝑒𝑒 𝑥𝑥
𝑒𝑒 𝑥𝑥 𝑒𝑒 ℎ − 𝑒𝑒 𝑥𝑥
𝑥𝑥 = lim
= lim
= lim
ℎ→0
ℎ→0
ℎ→0
ℎ
ℎ
𝑥𝑥
ℎ
𝑒𝑒 𝑥𝑥 (𝑒𝑒 ℎ −1)
𝑒𝑒
−1
= lim
= 𝑒𝑒 𝑥𝑥 lim
= 𝑒𝑒 𝑥𝑥 𝑓𝑓𝑓(0)
ℎ→0
ℎ→0
ℎ
ℎ
ℎ
0.1
𝑒𝑒 ℎ − 1
ℎ
1.71828
1.05171
0.01
1.00502
0.001
1.00050
0.0001
1.00005
0.00001
1.00001
By definition, this is derivative 𝑓𝑓𝑓(0) ,
what is the slope of 𝑒𝑒 𝑥𝑥 at 𝑥𝑥 = 1.
𝑒𝑒 ℎ − 1
lim
=1
ℎ→0
ℎ
𝑑𝑑 𝑥𝑥
𝑒𝑒 = 𝑒𝑒 𝑥𝑥
𝑑𝑑𝑑𝑑
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
𝑦𝑦 = 𝑎𝑎 𝑥𝑥
𝑑𝑑 𝑥𝑥
𝑎𝑎
𝑑𝑑𝑑𝑑
=
𝑑𝑑 𝑥𝑥
𝑎𝑎
𝑑𝑑𝑑𝑑
⟹
𝑑𝑑
𝑑𝑑𝑑𝑑
ln 𝑦𝑦 = ln 𝑎𝑎 𝑥𝑥 = 𝑥𝑥 ln 𝑎𝑎
(𝑒𝑒 𝑥𝑥 ln 𝑎𝑎 ) = 𝑒𝑒 𝑥𝑥 ln 𝑎𝑎
= 𝑎𝑎 𝑥𝑥 ∙ ln 𝑎𝑎
𝑑𝑑
𝑑𝑑𝑑𝑑
⟹ 𝑎𝑎 𝑥𝑥 = 𝑒𝑒 𝑥𝑥 ln 𝑎𝑎
a
(𝑥𝑥 ln 𝑎𝑎) = 𝑒𝑒 𝑥𝑥 ln 𝑎𝑎 ∙ ln 𝑎𝑎
x
𝑦𝑦 = ln 𝑥𝑥
𝑒𝑒 𝑦𝑦 = 𝑥𝑥 ⟹
𝑑𝑑 𝑦𝑦
𝑑𝑑
𝑒𝑒 =
𝑥𝑥
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
𝑒𝑒 𝑦𝑦
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
1
𝑑𝑑
ln 𝑥𝑥 =
𝑥𝑥
𝑑𝑑𝑑𝑑
= 1
⟹
⟹ 𝑥𝑥
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
= 1 ⟹
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
=
1
𝑥𝑥
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 𝑓𝑓(𝑥𝑥) = ln |𝑥𝑥|.
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 𝑥𝑥 𝑥𝑥 , 𝑥𝑥 sin 𝑥𝑥 … , , or a crazy
product, quotient, chain problem you do a simple trick:
FIRST find logarithm , 𝑙𝑙𝑙𝑙, 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:
𝑦𝑦 = 𝑥𝑥 sin 𝑥𝑥
𝑦𝑦 ′ =?
1 ′
sin 𝑥𝑥
ln 𝑦𝑦 = (sin 𝑥𝑥) ln 𝑥𝑥 ⇒
𝑦𝑦 = (cos 𝑥𝑥) ln 𝑥𝑥 +
𝑦𝑦
𝑥𝑥
𝑦𝑦 ′ = (ln 𝑥𝑥)𝑥𝑥 sin 𝑥𝑥 cos 𝑥𝑥 + (sin 𝑥𝑥) 𝑥𝑥 sin 𝑥𝑥−1
Try: 𝑦𝑦 = (sin 𝑥𝑥) 𝑥𝑥