ln = ln ln . a a b xy Example 1: Expand the logarithmic expression ln 4 . 2 ln + 4 ln ln 2 x y [ln The Number (Page 327) e = 1/u d u] [du/dx] = , > 0 u The base for the natural logarithm is defined using the fact dx that the natural logarithmic function is continuous, is one-to-one, and has a range of So, there must a unique real number x What you should How to understand the such that ln = 1 . This number is denoted by x definition of the number the letter , which has the decimal e e approximation 2.71828182846 . The Derivative of the Natural Logarithmic Function (Pages Let be a differentiable function of Complete the following u x. rules of differentiation for the natural logarithmic function: What you should learn [ln How to find derivatives = 1/x , > 0 d x] x of functions involving the dx natural logarithmic function Example 2: Find the derivative of (x) = ln . f x2 x + 2x lnx x If is a differentiable function of such that 0 , then u x u [ln ] = . In trial words, functions of d u dx ln the form y = can be differentiated as if the absolute u value signs were not present . Homework Assignment Page(s) Exercises
ln a = ln a ln b . Example 1: Expand the logarithmic expression ln ln x + 4 ln y ln 2 xy 4 . 2 The Number e (Page 327) The base for the natural logarithm is defined using the fact that the natural logarithmic function is continuous, is one-to-one, and has a range of So, there must a unique real number x such that ln x = 1 . This number is denoted by the letter e , which has the decimal approximation 2.71828182846 . The Derivative of the Natural Logarithmic Function (Pages Let u be a differentiable function of x. Complete the following rules of differentiation for the natural logarithmic function: d [ln x] = 1/x , x > 0 dx d [ln u] = 1/u [du/dx] = , u > 0 dx What you should How to understand the definition of the number e What you should learn How to find derivatives of functions involving the natural logarithmic function Example 2: Find the derivative of x + 2x lnx f (x) = x2 ln x . If u is a differentiable function of x such that u 0 , then d [ln u ] = . In words, functions of dx the form y = ln u can be differentiated as if the absolute value signs were not present . Homework Assignment Page(s) Exercises trial
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