Universiti Pendidikan Sultan Idris

1. Beginning Calculus
- Derivatives of Exponential and Logarithmic Functions -
Shahrizal Shamsuddin Norashiqin Mohd Idrus
Department of Mathematics,
FSMT - UPSI
(LECTURE SLIDES SERIES)
VillaRINO DoMath, FSMT-UPSI
(D6) Derivatives of Exponential and Logarithmic Functions 1 / 16

2. Derivatives of Exponential Functions Derivatives of Logarithmic Functions
Learning Outcomes
Compute the derivatives of exponential functions.
Compute the derivatives of logarithmic functions.
VillaRINO DoMath, FSMT-UPSI
(D6) Derivatives of Exponential and Logarithmic Functions 2 / 16

3. Derivatives of Exponential Functions Derivatives of Logarithmic Functions
y = ax where a, x 2 R. a is called the base with a > 0 but a 6= 1.
Natural exponential function y = ex with e 2.71828 . . . .
a is …xed and x varies.
a0 = 1, a1 = a, an = a a a a| {z }
n times
Some rules of exponents:
am+n = am an
(am )n
= amn
am/n = n
p
am
VillaRINO DoMath, FSMT-UPSI
(D6) Derivatives of Exponential and Logarithmic Functions 3 / 16

4. Derivatives of Exponential Functions Derivatives of Logarithmic Functions
The graph of y = 2x .
-4 -2 0 2 4
2
4
x
y x y
...
...
1
1
2
0 1
1 2
...
...
VillaRINO DoMath, FSMT-UPSI
(D6) Derivatives of Exponential and Logarithmic Functions 4 / 16

5. Derivatives of Exponential Functions Derivatives of Logarithmic Functions
Derivative of a^x
d
dx
(ax
) = ax
ln x (1)
VillaRINO DoMath, FSMT-UPSI
(D6) Derivatives of Exponential and Logarithmic Functions 5 / 16

6. Derivatives of Exponential Functions Derivatives of Logarithmic Functions
Exponent and Logarithm
y = loga x where a, x 2 R with a > 1 and x > 0. Natural logarithmic
function y = ln x.
Relationship between exponents and logarithms
y = ex
, ln y = x (2)
Some rules of logarithm:
ln (m n) = ln m + ln n
ln 1 = 0; ln e = 1
VillaRINO DoMath, FSMT-UPSI
(D6) Derivatives of Exponential and Logarithmic Functions 6 / 16

7. Derivatives of Exponential Functions Derivatives of Logarithmic Functions
Graphs of Exponent and Logarithm
Graphs of y = ex and y = ln x.
-4 -2 2 4
-4
-2
2
4
x
y
VillaRINO DoMath, FSMT-UPSI
(D6) Derivatives of Exponential and Logarithmic Functions 7 / 16

8. Derivatives of Exponential Functions Derivatives of Logarithmic Functions
Derivative of ln x
d
dx
(ln x) =
1
x
(3)
VillaRINO DoMath, FSMT-UPSI
(D6) Derivatives of Exponential and Logarithmic Functions 8 / 16

9. Derivatives of Exponential Functions Derivatives of Logarithmic Functions
Example
Let y = xx .
dy
dx
= xx (ln x + 1)
VillaRINO DoMath, FSMT-UPSI
(D6) Derivatives of Exponential and Logarithmic Functions 9 / 16

10. Derivatives of Exponential Functions Derivatives of Logarithmic Functions
To get e.
Evaluate lim
n!∞
1 +
1
n
n
.
Take natural log.
ln 1 +
1
n
n
= n ln 1 +
1
n
Let ∆x =
1
n
. ∆x ! 0 as n ! ∞. Then,
ln 1 +
1
n
n
= n ln 1 +
1
n
=
1
∆x
ln (1 + ∆x) ln 1
=
ln (1 + ∆x) ln 1
∆x
VillaRINO DoMath, FSMT-UPSI
(D6) Derivatives of Exponential and Logarithmic Functions 10 / 16

11. Derivatives of Exponential Functions Derivatives of Logarithmic Functions
Continue
Take limits.
lim
n!∞
ln 1 +
1
n
n
= lim
∆x!0
ln (1 + ∆x) ln 1
∆x
=
d
dx
ln x
x=1
= 1
So,
lim
n!∞
1 +
1
n
n
= e
lim
n!∞
ln 1+
1
n
!n
= e1
= e
By taking n ! ∞ (large values of n ), will get closer to the value of
e.
VillaRINO DoMath, FSMT-UPSI
(D6) Derivatives of Exponential and Logarithmic Functions 11 / 16

12. Derivatives of Exponential Functions Derivatives of Logarithmic Functions
Example
d
dx
ln x2
d
dx
ln x2
=
1
x2
d
dx
x2
=
2x
x2
=
2
x
In general, if u is any function. Then,
(ln u)0
=
u0
u
VillaRINO DoMath, FSMT-UPSI
(D6) Derivatives of Exponential and Logarithmic Functions 12 / 16

13. Derivatives of Exponential Functions Derivatives of Logarithmic Functions
Example
d
dx
[ln (sec x)] = tan x
VillaRINO DoMath, FSMT-UPSI
(D6) Derivatives of Exponential and Logarithmic Functions 13 / 16

14. Derivatives of Exponential Functions Derivatives of Logarithmic Functions
Example
d
dx
ex tan 1 x = ex tan 1 x tan 1 x +
1
1 + x2
VillaRINO DoMath, FSMT-UPSI
(D6) Derivatives of Exponential and Logarithmic Functions 14 / 16

15. Derivatives of Exponential Functions Derivatives of Logarithmic Functions
The Derivative of log u (base a)
d
dx
(loga u) =
d
dx
ln u
ln a
=
1
ln a
d
dx
(ln u)
=
1
ln a
u0
u
VillaRINO DoMath, FSMT-UPSI
(D6) Derivatives of Exponential and Logarithmic Functions 15 / 16

16. Derivatives of Exponential Functions Derivatives of Logarithmic Functions
The Power Rule - for any Real n.
If f (x) = xr , with r 2 R, then
f 0
(x) = rxr 1
(4)
Proof: Use natural exponential and logarithm (with x = eln x )
xr
= eln xr
d
dx
(xr
) =
d
dx
eln xr
=
d
dx
er ln x d
dx
(r ln x)
= er ln x r
x
= xr r
x
= rxr 1
VillaRINO DoMath, FSMT-UPSI
(D6) Derivatives of Exponential and Logarithmic Functions 16 / 16