Derivative of ln(x)
and exp(x)
We will learn how to differentiate functions involving
ln(x) and ex. The two basic
rules are:
If y = ex then y0 = ex
1
0
If y = ln(x) then y =
x
Use these rules in combination with the other rules and
the simplifying ln( ) rules.
Example
Q?
Take the derivative
of the function y = ln(x)(sec x)
Example
Q?
Take the derivative
of the function y = ln(sec x)
Example
Q?
Take the derivative
of the function
ex
y= 2
x −x
Example
Q?
Take the derivative
2
of the function
2
x
y = e −x
We can also use the log
rules on a question first to
simplify a question before we
differentiate.
Example
Q?
Take the derivative
of
"
e2x
y = ln
sin(x) cos(x)
A.
#
First use the log
3
rules to "simplify.
#
e2x
y = ln
sin(x) cos(x)
h i
y = ln e2x − ln [sin(x) cos(x)]
y = 2x − ln [sin(x)] − ln [cos(x)]
Now differentiate.
1
1
(− cos(x)) −
(sin(x))
sin(x)
cos(x)
y 0 = 2 + cot(x) − tan(x)
y0 = 2 −
See how by simplifying the
question first avoided a lot of
chain rule and the quotient rule
and the exponential function.
The differentiating was much
easier. We could use the log
4
rules because the ln() was
present but we could introduce
it ourselves if we choose.
Example
Q?
Differentiate
x2(x − 3)33
y=
x−2
A.
First take the ln() of
both sides.
#
"
x2(x − 3)33
ln(y) = ln
x−2
h i
h
i
= ln x2 + ln (x − 3)33 − ln [x − 2]
ln(y) = 2 ln(x) + 33 ln(x − 3) − ln(x − 2)
Now differentiate implicitly
33
1
1 0 2
y = +
−
y
x x−3 x−2
5
Solve for ·y.
2
0
+
y =y
x
33
1
−
x−3 x−2
¸
Replace
y as a function
of x.
#
"
y0 =
·
2
33
x (x − 3)
2
x−2
33
1
+
−
x x−3 x−2
The technique illustrated in
the above Example is called
Logarithmic Differentiation.
Now we are ready to support our claim that e is a special
number. First another example
of logarithmic differentiation.
Example
Q?
Differentiate y = bx
6
¸
A.
Use logarithmic
differentiation.
y = bx
ln(y) = ln(bx)
ln(y) = x ln(b)
1 0
y = ln(b)
y
y 0 = y ln(b)
y 0 = bx ln(b)
In Example above, if b = e
then y = y0 = ex since ln(e) = 1.
See how special e is? Impressed?
What if the function I need
to differentiate has a logarithm
7
with a base other than e? Use
the change of base formula to
change it into base e first, then
differentiate.
Example
Q?
Differentiate y =
log(x)
A.
base.
First change the
y = log(x)
ln(x)
=
ln(10)
Now differentiate, noting that
ln(10) is a constant.
y0 =
1
x ln(10)
8
Ok, practice, practice, practice.
0.0.1 Homework
Section 3.1 # 8, 9, 20, 21, 34,
43
Submit 8, 20, 34
Section 3.2 # 3 - 6, 10, 13,
14, 16, 19, 35, 36, 41
Submit # 4, 6, 10, 14, 36,
41
Section 3.5 # 7 - 32, 51, 52
Submit 8, 10, 12, 14, 16,
18, 20, 22, 24, 32, 52
9