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CHAPTER 2
2.4 Continuity
Derivatives
of Logarithmic
Functions
( d / dx ) (log a x) = 1 / ( x ln a )
( d / dx ) (ln x) = 1 / x
( d / dx ) (ln u) = (1 / u) ( du / dx )
( d / dx ) [ln g(x)] = g’(x) / g(x)
( d / dx ) ln |x| = 1 / x
Example Differentiate
the
functions.
CHAPTER 2
a) f (x) = ln (2 – x )
2.4 Continuity
b) f (x) = log [x / (x – 1)]
Example Differentiate
f
and
find
its
CHAPTER 2
domain for f (x) = ln ln x.
2.4 Continuity
Steps in Logarithmic Differentiation
1. Take natural logarithms of both sides
of an equation y = f (x) and use the
Laws of Logarithms to simplify.
2. Differentiate implicitly with respect
to x.
3. Solve the resulting equation for y’.
Power Rule If n is any real number and
f (x) = x n, then f’ (x) = n x n –1 .
You should distinguish carefully
between the Power Rule, where the base
is variable and the exponent is constant,
and the rule for differentiating
exponential functions, where the base is
constant and the exponent is variable.
In general, there are 4 cases for exponents
and bases:
1. d /dx (a b) = 0 ( a and b are constants)
2. d /dx [ f (x)b] = b [ f (x)]b-1 f’(x)
3. d /dx (a g(x)) = a g(x) (ln a) g’(x)
4. To find (d / dx) [ f(x)]g(x), logarithmic
differentiation can be used.
Example Differentiate y = x x .
The number e as a Limit
lim x 0 ( 1 + x ) 1 / x = e
Example Show that
lim n 00 ( 1 + ( x / n )n = e x for any x>0.