Presentation on theme: "Derivatives of Logarithmic Functions"— Presentation transcript:
1 Derivatives of Logarithmic Functions CHAPTER 22.4 ContinuityDerivatives 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 Example Differentiate the functions a) f (x) = ln (2 – x ) b) f (x) = log [x / (x – 1)]CHAPTER 22.4 Continuity
4 Example Differentiate f and find its domain for f (x) = ln ln x. CHAPTER 22.4 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 carefullybetween the Power Rule, where the baseis variable and the exponent is constant,and the rule for differentiatingexponential functions, where the base isconstant 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.