By Bernard V Zandy; Jonathan J White
Calculus : CliffsQuickReview Calculus, Anton/Bivens/Davis model by way of Cliffs speedy assessment Publishing employees. Cliffs Notes, Inc.,2003
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1, then f l(x) = ( ln a) $ x Note that the exponential function f (x) = ex has the special property that its derivative is the function itself, f '(x) = ex = f (x). Example 3-28: Find f l(x) if f (x) = e x 2 +5 2 . +5 $ 2x f l(x) = 2x $ e x2 + 5 f l(x) = e x Example 3-29: Find y l if y = 5 x . y l= ( ln 5) $ 5 $ 12 x - 1/2 = ( ln 5) $ 5 $ 1 2 x x x ( ln 5) $ 5 $ 2 x x y l= 42 CliffsQuickReview Calculus Example 3-30: Find f'(x) if f (x) = 1n(sin x). 1 $ cos x f l(x) = sin x cos = sin xx f l(x) = cot x dy Example 3-31: Find dx if y = log10 (4x 2 - 3x - 5).
1+x (5) If f (x) = sec - 1 x = arc sec x, 0 # f (x) # r, f (x) ! r 2 then 1 f l(x) = . x x2- 1 r (6) If f (x) = csc - 1 x = arc csc x, - r 2 # f (x) # 2 , f (x) ! 0 -1 then f l(x) = . x x2- 1 Example 3-26: Find f '(x) if f (x) = cos-1(5x). -1 $5 1 - (5x) 2 -5 f l(x) = 1 - 25x 2 f l(x) = Chapter 3: The Derivative 41 Example 3-27: Find y l if y = arctana x 3 k . Because y = arctan(x 3/2 ) 1 $ 3 x 1/2 1 + (x 3/2 ) 2 2 = 1 3 $ 32 x 1/2 1+x y l= y l= 3 x 2 (1 + x 3 ) Differentiation of Exponential and Logarithmic Functions Exponential functions and their corresponding inverse functions, called logarithmic functions, have the following differentiation formulas: (1) If f (x) = e x , then f l(x) = e x .
The largest function value from the previous step is the maximum value, and the smallest function value is the minimum value of the function on the given interval. Example 4-5: Find the maximum and minimum values of f (x) = sin x + cos x on [0,2π]. The function is continuous on [0,2π], and from Example 4-4, the critcal points are a r/4, 2 k and a5r/4, - 2 k. The function values at the end points of the interval are f (0)=1 and f (2π)=1; hence, the maximum function value of f (x) is 2 at x=π/4, and the minimum function value of f (x) is - 2 at x = 5π/4.
CliffsQuickReview calculus : Anton/Bivens/Davis version by Bernard V Zandy; Jonathan J White