4.4
Differentiation of
()Suppose
() = () where () is a differentiable function. Then
ln () = ln () = () Thus, (ln ()) = 0() Since (ln ()) = 0() () So 0() () = 0() this implies 0() = 0() () = 0() () Theorem 57 Let be a differentiable function of
1 [ ] = 2 [ ] =
Example 110 Find the derivative of () = −2+6−9 Example 111 Find the derivative of = 70−002
Example 112 Find the derivative of () = 2+3
Example 113 Find the relative extrema of () =
1.25 0 -1.25 -2.5 12.5 10 7.5 5 2.5 0 x y x y 61
Example 114 Sketch the graph of () = 12−002 for ≥ 0 12−002 500 375 250 125 0 250 200 150 100 50 0 x y x y
Theorem 58 If () is a polynomial function and 0 lim
→∞ ()
= 0
Proof. By L’Hopitai’s rule. Theorem 59 Suppose
0() = ()
where 6= 0 and () 0 Then () is an exponential function that can be expressed in the form
() =
Exercise 19 1,3,6,8,10,12,13,16,17,20,22,23,24,25,26,29,31,33,35.
