This document discusses the derivatives of exponential and logarithmic functions. It provides formulas for computing the derivatives of exponential functions like y = ax and natural exponential functions. It also gives formulas for computing the derivatives of logarithmic functions like y = loga(x) and natural logarithmic functions. Examples are worked through demonstrating how to apply these formulas to find the derivatives of various exponential and logarithmic expressions. The key formulas presented are the derivatives of ax, ln(x), and loga(x).

1. Beginning Calculus
- Derivatives of Exponential and Logarithmic Functions -
Shahrizal Shamsuddin Norashiqin Mohd Idrus
Department of Mathematics,
FSMT - UPSI
(LECTURE SLIDES SERIES)
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2. Derivatives of Exponential Functions Derivatives of Logarithmic Functions
Learning Outcomes
Compute the derivatives of exponential functions.
Compute the derivatives of logarithmic functions.
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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
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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
...
...
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5. Derivatives of Exponential Functions Derivatives of Logarithmic Functions
Derivative of a^x
d
dx
(ax
) = ax
ln x (1)
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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
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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
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8. Derivatives of Exponential Functions Derivatives of Logarithmic Functions
Derivative of ln x
d
dx
(ln x) =
1
x
(3)
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9. Derivatives of Exponential Functions Derivatives of Logarithmic Functions
Example
Let y = xx .
dy
dx
= xx (ln x + 1)
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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
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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.
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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
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13. Derivatives of Exponential Functions Derivatives of Logarithmic Functions
Example
d
dx
[ln (sec x)] = tan x
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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
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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
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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
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