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section6.1 - Math 2B Dr C Famiglietti Section 6.1 The...

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Math 2 B Dr. C. Famiglietti ************************************************************************ Section 6.1: The Natural Logarithm y = ln x is the natural logarithmic function (which means e y = x , where e is a constant approximately equal to 2.71828…). The graph of y = ln x is shown below and has the following characteristics: Domain: 0, ( ) Range: −∞ , ( ) Increasing throughout its domain Concave downward Passes through 1,0 ( ) since ln1 = 0 lim x →∞ ln x = lim x 0 + ln x = −∞ Properties of the natural logarithmic function ( a and b are real numbers greater than 0 and r is a rational number) 1. ln1 = 0 2. ln e = 1 3. ln ab ( ) = ln a + ln b 4. ln a b = ln a ln b 5. ln a r ( ) = r ln a Derivative of the natural logarithmic function d dx ln x ( ) = 1 x Combine with the Chain Rule to obtain: d dx ln u ( ) = 1 u du dx , where u = g x ( ) Corresponding antiderivative 1 x dx = ln x + c
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Logarithmic differentiation (a technique of differentiation which is used to differentiate functions of the form f x ( ) [ ] g x ( ) and is helpful when differentiating functions with multiple products, quotients and/or exponents) 1. Write the equation with y on the left and all x ’s on the right.
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