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CHAPTER 2

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.

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CHAPTER 2

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  1. CHAPTER 2 2.4 Continuity Derivatives of Logarithmic Functions

  2. ( 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

  3. Example Differentiate the functions. a) f (x) = ln (2 – x ) b) f (x) = log [x / (x – 1)] CHAPTER 2 2.4 Continuity

  4. Example Differentiate f and find its domain for f (x) = ln ln x. CHAPTER 2 2.4 Continuity

  5. Steps in Logarithmic Differentiation • Take natural logarithms of both sides of an equation y = f (x) and use the Laws of Logarithms to simplify. • Differentiate implicitly with respect to x. • Solve the resulting equation for y’.

  6. 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.

  7. 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.

  8. Example Differentiate y = x x .

  9. The number e as a Limit lim x  0 ( 1 + x ) 1 /x = e

  10. Example Show that lim n  00 ( 1 + ( x / n )n = e xfor any x>0.

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