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(𝑥) is an increasing function.
There are basically three kinds of exponential functions 𝑦 = 𝑎𝑥.
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
limSo, how to
=1
ℎ→0
ℎ
calculate the slope
when you don’t
know derivative?
𝑑 𝑥
𝑒 = 𝑒𝑥
𝑑𝑥
example: Differentiate the function 𝑦 = 𝑒 tan 𝑥
𝑢 = tan 𝑥
𝑑𝑦 𝑑𝑦 𝑑𝑢
𝑑𝑢
=
= 𝑒𝑢
= 𝑒 tan 𝑥 𝑠𝑒𝑐 2 𝑥
𝑑𝑥 𝑑𝑢 𝑑𝑥
𝑑𝑥
example:
Find 𝑦’ if 𝑦 = 𝑒 −4𝑥 sin 5𝑥.
𝑦 ′ = 𝑒 −4𝑥 (−4)(sin 5𝑥) + 𝑒 −4𝑥 (cos 5𝑥)(5)
𝑦 ′ = 𝑒 −4𝑥 (5 cos 5𝑥 − 4 sin 5𝑥)
Derivative of exponential function 𝑓 𝑥 =
𝑎 𝑥 = 𝑒 𝑥 ln 𝑎
a
x
?
𝑎 𝑥 = 𝑒 𝑥 ln 𝑎
take ln of both sides
𝑥 ln 𝑎 = 𝑥 ln 𝑎 (ln 𝑒)
take derivative of both sides
𝑑 𝑥
𝑎
𝑑𝑥
= 𝑒 𝑥 ln 𝑎 ∙ ln 𝑎
𝑑 𝑥
𝑎
𝑑𝑥
= 𝑎 𝑥 ∙ ln 𝑎
𝑑 𝑢
𝑑𝑢
(𝑒 ) = 𝑒 𝑢
𝑑𝑥
𝑑𝑥
Derivative of natural logarithm function 𝑓 𝑥 = ln 𝑥
𝑦 = ln 𝑥
𝑒𝑦 = 𝑥
𝑑 𝑦
𝑑
𝑒 =
𝑥
𝑑𝑥
𝑑𝑥
𝑒𝑦
𝑥
𝑑𝑦
= 1
𝑑𝑥
𝑑𝑦
= 1
𝑑𝑥
𝑑𝑦 1
=
𝑑𝑥 𝑥
𝑑
1
ln 𝑥 =
𝑑𝑥
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.
LOGARITHMIC DIFFERENTIATION
1.
example: differentiate
Take natural logarithms of both sides
of an equation y = f(x) and simplify.
2. Differentiate implicitly with respect to x.
3. Solve the resulting equation for y’.
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 ′ =?
ln 𝑦 = (sin 𝑥) ln 𝑥
1 ′
sin 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