benginning calculus lecture notes 7 - exp, log
TRANSCRIPT
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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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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Derivatives of Exponential Functions Derivatives of Logarithmic Functions
y = ax where a, x ∈ R. a is called the base with a > 0 but a 6= 1.Natural exponential function y = ex with e ≈ 2.71828 . . . .
a is fixed and x varies.
a0 = 1, a1 = a, an = a · a · a · · · · · a︸ ︷︷ ︸n times
Some rules of exponents:
am+n = am · an(am)n = amn
am/n = n√am
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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 12
0 11 2...
...
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Derivatives of Exponential Functions Derivatives of Logarithmic Functions
Derivative of a^x
ddx(ax ) = ax ln x (1)
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Derivatives of Exponential Functions Derivatives of Logarithmic Functions
Exponent and Logarithm
y = loga x where a, x ∈ R with a > 1 and x > 0. Natural logarithmicfunction y = ln x .
Relationship between exponents and logarithms
y = ex ⇔ ln y = x (2)
Some rules of logarithm:
ln (m · n) = lnm + ln nln 1 = 0; ln e = 1
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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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Derivatives of Exponential Functions Derivatives of Logarithmic Functions
Derivative of ln x
ddx(ln x) =
1x
(3)
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Derivatives of Exponential Functions Derivatives of Logarithmic Functions
Example
Let y = xx .
dydx= xx (ln x + 1)
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Derivatives of Exponential Functions Derivatives of Logarithmic Functions
To get e.
Evaluate limn→∞
(1+
1n
)n.
Take natural log.
ln(1+
1n
)n= n ln
(1+
1n
)
Let ∆x =1n. ∆x −→ 0 as n −→ ∞. Then,
ln(1+
1n
)n= n ln
(1+
1n
)=
1∆x
ln (1+ ∆x)− ln 1
=ln (1+ ∆x)− ln 1
∆x
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Derivatives of Exponential Functions Derivatives of Logarithmic Functions
Continue
Take limits.
limn→∞
ln(1+
1n
)n= lim
∆x→0ln (1+ ∆x)− ln 1
∆x=
ddxln x
∣∣∣∣x=1
= 1
So,
limn→∞
(1+
1n
)n= e
limn→∞
ln
(1+1n
)n
= e1
= e
By taking n→ ∞ (large values of n ), will get closer to the value ofe.
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Derivatives of Exponential Functions Derivatives of Logarithmic Functions
Example
ddx
(ln x2
)ddx
(ln x2
)=
1x2· ddx
(x2)
=2xx2=2x
In general, if u is any function. Then,
(ln u)′ =u′
u
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Derivatives of Exponential Functions Derivatives of Logarithmic Functions
Example
ddx[ln (sec x)] = tan x
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Derivatives of Exponential Functions Derivatives of Logarithmic Functions
Example
ddx
(ex tan
−1 x)=(ex tan
−1 x)(
tan−1 x +1
1+ x2
)
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Derivatives of Exponential Functions Derivatives of Logarithmic Functions
The Derivative of log u (base a)
ddx(loga u) =
ddx
(ln uln a
)=
1ln a
ddx(ln u)
=1ln a· u′
u
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Derivatives of Exponential Functions Derivatives of Logarithmic Functions
The Power Rule - for any Real n.
If f (x) = x r , with r ∈ R, then
f ′ (x) = rx r−1 (4)
Proof: Use natural exponential and logarithm (with x = e ln x )
x r = e ln xr
ddx(x r ) =
ddx
(e ln x
r)
=ddx
(er ln x
)· ddx(r ln x)
= er ln x · rx
= x r · rx= rx r−1
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