The exponential function occurs very frequently in mathematical models of nature and society.
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Exponential Functions If is a continuous function with domain R and range (0, ∞). In particular, for all . If is a decreasing function. If is an increasing function. There are basically three kinds of exponential functions . The exponential function occurs very frequently in mathematical models of nature and society.
Transcript of The exponential function occurs very frequently in mathematical models of nature and society.
Exponential Functions
If is a continuous function with domain R and range (0, ∞).
In particular, for all .
If is a decreasing function.
If is an increasing function.
There are basically three kinds of exponential functions .
The exponential function occurs very frequently in mathematical models of nature and society.
Derivative of exponential function
By definition, this is derivative , what is the slope of at .
𝑓 ′ (𝑥 )=limh→0
𝑓 (𝑥+h )− 𝑓 (𝑥 )𝑥
=limh→0
𝑒𝑥+h−𝑒𝑥
h=limh→0
𝑒𝑥𝑒h−𝑒𝑥
h
¿ limh→0
𝑒𝑥 (𝑒¿¿h−1)h
=𝑒𝑥 limh→0
𝑒h−1h
=𝑒𝑥 𝑓 ′ (0)¿
1 1.71828
0.1 1.05171
0.01 1.00502
0.001 1.00050
0.0001 1.00005
0.00001 1.00001
limh→0
𝑒h−1h
=1
𝑑𝑑𝑥
𝑒𝑥
=𝑒𝑥
So, how to calculate the slope when you don’t know derivative?
example: Differentiate the function 𝑦=𝑒 tan 𝑥𝑢=tan 𝑥
example:
Find
if .
𝑑𝑦𝑑𝑥
=𝑑𝑦𝑑𝑢
𝑑𝑢𝑑𝑥
=𝑒𝑢𝑑𝑢𝑑𝑥
=𝑒tan 𝑥 𝑠𝑒𝑐2 𝑥
𝑦 ′=𝑒−4 𝑥(−4)¿
𝑦 ′=𝑒−4 𝑥¿
𝑎𝑥=𝑒𝑥 ln𝑎
Derivative of exponential function
xa
𝑑𝑑𝑥
(𝑒¿¿𝑢)=𝑒𝑢𝑑𝑢𝑑𝑥
¿
take derivative of both sides
take ln of both sides𝑎𝑥=𝑒𝑥 ln𝑎
𝑥 ln𝑎=𝑥 ln𝑎 ¿¿
?
𝑦=ln𝑥
𝑒𝑦 𝑑𝑦𝑑𝑥
=1
Derivative of natural logarithm function
𝑑𝑑𝑥
𝑒𝑦=𝑑𝑑𝑥
𝑥
𝑥𝑑𝑦𝑑𝑥
=1
𝑑 𝑦𝑑𝑥
=1𝑥
𝑑𝑑𝑥
( ln𝑥 )= 1𝑥
Differentiate y = ln(x3 + 1).
example:
To use the Chain Rule, we let u = x3 + 1.
Then, y = ln u.
example:
Find:
Differentiateexample: ( ) lnf x x
example:
If we first simplify the given function using the laws of logarithms, the differentiation becomes easier
example:
Find f ’(x) if
Thus, f ’(x) = 1/x for all x ≠ 0.
The result is worth remembering:
Derivative of Logarithm Function
a logarithmic function with base a in terms of the natural logarithmic function:
Since 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, or a crazy product, quotient, chain problem you do a simple trick:FIRST find logarithm so you’ll have sum instead of product, and product instead of exponent. Life will be much, much easier.
LOGARITHMIC DIFFERENTIATION
1. 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’.
example: differentiate
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.
𝑦=𝑥sin 𝑥 𝑦 ′=?
ln 𝑦=¿¿¿
𝑦 ′=¿
Try:
example:
1𝑦𝑦 ′=¿
