UL1: Review o Prerequisite Skills or Unit # (Derivatives) Working with the properties o exponents Simpliying radical expressions Finding the slopes o parallel and perpendicular lines Simpliying rational expressions. (add/subtract/multiply/divide and restrictions) (HANDOUT) Expanding and actoring algebraic expressions Evaluating Expressions Working with the dierence quotient. Homework: p. 6 # 1-11 UL: Sec..1 The Derivative Function In the previous chapter we discussed the slope o a tangent line and rates o change using limits. ( x + h) ( x) lim h 0 h This limit has two interpretations: the slope o the tangent and the instantaneous rate o change. Limits are used so oten in calculus they have be given a speciic name and concise notation. The limit is called the Several accepted notations o derivatives are; N.B. is itsel a unction and it equals the slope o the unction at the point ( x, (x) ). Eg.1: Find the derivative o ( ) x = x @ x = -4. (using irst principles) [ sol: -8 ] I we want to ind the derivative o a unction at numerous points it is tedious i we use the deinition each time. So, we use the deinition but keep the variable x instead o a speciic number. Page 1 o 11
Eg.: The derivative o a unction at an arbitrary value. Find the derivative o ( x) = x. (using irst principles) [ (x) = x ] Now solve or (-11), (0), (5). N.B. Notice that the derivative o a quadratic unction (degree two) is a linear unction (degree one). Complete Investigation p.67 as a class. A) Determine the derivative with respect to x o each o the ollowing unctions: 4 5 i) ( x) = x ii) ( x) = x iii) ( x) = x iv) ( x) = x B) What pattern do you see developing. x = x C) Predict the derivative o ( ) 9 D) Generalize the derivative or ( ) n The Deinition o the Derivative Function x = x, when n is a positive integer? The derivative o (x) with respect to x is the unction (x) where ( x) this limit exists. ( x + h) ( x) ' = lim, provided that h 0 h Page o 11
We can now deine (instantaneous velocity) as the derivative o the position unction with respect to time. I the position unction at time t is given by s(t), then the velocity at time t is given by; The process o inding the derivative o a unction is called dierentiation. A unction is dierentiable at a i (a) exists. n.b. I you can not draw a unique, deined tangent at a then the unction is not dierentiable at a. ways derivatives ail to exist: The Normal o a Tangent: Homework: (p. 7 # 1, 5, 6d, 7b, 9, 10, 1cd, 15, 19 ) Page o 11
UL: Sec.. The Derivatives o Polynomial Functions We have seen that derivatives o unctions are o practical use because they represent instantaneous rates o change. I we have to compute derivatives using irst principles all the time then it is a very tedious and timeconsuming process. Let s look at some rules that will make this process easier. 1. The Constant Function Rule I (x) = k, where k is a constant, then (x) = 0. Since the graph o any constant unction is a horizontal line with slope zero at each point, the derivative should be zero. Ie: ( x) =, '( x) = 0. The Power Rule I (x) = x n, where n is a real number, then (x) = nx n-1. In the previous sections investigation we discovered this. x = x then x = x then x Ie: ( ), '( ) = ( ) 5, ' ( ) = g ( x) = x, g( x) =, g '( x) =, g '( x) = x Eg.1: Using the power rule. a) ( x) = x then ' 1, ( x) = b) ( x) g = x then g ' 1 5 or x 5 ( x) =, Page 4 o 11
. The Constant Multiple Rule I ( x) = k g( x), where k is any constant, then '( x) k g '( x) =. Eg.: Using the constant multiple rule. a) ( x) = 1x then b) ( x) 18x d ' = ( x) 1 ( x ) = 1 ( x ) 6x = then = ( ) 4. The Sum and Dierence Rule 4 4 1 d 4 ' x = 18 x = 18 x = 4x I the unctions p(x) and q(x) are dierentiable, and ( x) = p( x) ± q( x) then '( x) p '( x) ± q '( x) =. 1 Eg.: I ( x) = 7x 4x + then '( x) = Homework: p. 8 #,, 4, 6, 9bd, 10, 11, 1, 15, 0, 4** Page 5 o 11
UL4: Sec.. The Product Rule Since the derivative o a sum or dierence o unctions is simply the sum or dierence o their individual derivatives, you might assume that the derivative o a product o unctions is the product o their individual derivatives. This is not true. Eg.1: Let p( x) = ( x) g( x) where ( x) = x 1 and g ( x) = x + 8, show that p '( x) '( x) g '( x) The expression or p(x) can be simpliied to The Product Rule The derivative o the product o two unctions is the derivative o the irst unction times the second plus the derivative o the second unction times the irst. I p ( x) = ( x) g( x) then p' ( x) = Eg.1: Find h (x) or h( x) ( x + )( 4 x ) = Sol: -10x 4-18x +8x Page 6 o 11
Eg.: Find g (x) or ( x) = ( 6x + 4x )( 4x + x + 7) g Sol: 10x 4 +11x +150x + x 6 The Extended Product Rule Find an expression or p '( x) i p( x) = ( x) g( x) h( x) The power o a unction rule or Positive Integers n n 1 I u is a unction o x, and n is a positive integer, then ( u) = n ( u) ( u) d d n or '( x) = n[ g( x) ] 1 g '( x) Eg.: a) Find the value o dy or x = - i ( ) y 4 = x x + 7 b) Find the value o dy or x = -1 i ( ) 5 ( ) 4 y = x + 1 x + Homework: p. 90 # 1cde, bc, 5ae, 6, 7, 8b, 1, 14 Page 7 o 11
Homework: p. 9/9 # 1-18 UL5: Mid-Chapter Review (Work Period) UL6: Sec..4 The Quotient Rule The product rule allows us to take the derivative o the product o two unctions. I h ( x) ( x) ( x) ( x) g( x) ( x) g '( x) [ g( x) ] ' = then h '( x) = where g x g ( ) 0 In words this is the derivative o the top times the bottom minus the derivative o the bottom times the top, all over the bottom squared. Eg.1: ( x) 6x = x 1 Eg.: ( x) 5 = x + 6 Page 8 o 11
Eg.: ( x) ( x + 1)( x + ) ( x 1)( x ) = @ x = 4 Homework: p. 97 # 4bc, 5cd, 6, 7, 8, 9b, 11, 1, 14, 17 UL7: Sec..5 The Derivatives o Composite Functions We have seen many ways to dierentiate combinations o polynomial unctions: I h h ( x) = p( x) ± q( x) then '( x) = p '( x) ± q '( x) ( x) = ( x) g( x) h' ( x) = '( x) g( x) + ( x) g '( x) ( ) ( x) ' ( ) ( x) g( x) ( x) g '( x) x = h' x = g( x) [ g( x) ] Another way that unctions are oten combined is called composition. In this case, one unction is substituted or another. You can think o a composite unction as an input-output diagram, when the output o one unction is used as the input or a second. input ( x) unction g g( x) unction output ( g( x) ) The new unction, (g(x)) is called the composition unction o and g, and is written g. Eg.1: I (x) = x and g(x) = 5x + x, then ind the unctions Sol: a) g( x) = ( g( x) ) = b) g ( x) = g( ( x) ) = c) g g( x) = g and g and g g. To ind the derivative o a composite unction ( x) ( g( x) ) h = we use the chain rule. Page 9 o 11
The Chain Rule I and g are unctions having derivatives, then the composite unction ( x) ( g( x) ) h = has a derivative given by: Work rom the outside to the inside. In words we say the derivative o the outer unction evaluated at the inner unction times the derivative o the inner unction. In Liebniz Notation: Eg.: I ( x) = x + h, ind h (x) 8 dy = ind. Eg.: I y ( x x + ) 10 5 Eg.4: I y = u + u + where u dy = 1 x, ind @ x = 1. Sol: Using the Leibniz Notation, dy = dy du du = 9 4 ( 10u + 5u ) ( 6x) It is not necessary to write this expression entirely in terms o x. Note that when x = 1 we have u = -. dy Thereore, = 040 Page 10 o 11 Homework: p. 105 # 1 de, 4, 8ac, 10, 1abd, 16, 17b
UL8: Implicit Dierentiation (p.564) + 4 An explicit unction is one that is expressed in terms o y (by itsel) ie: y = x. An implicit unction is when y is not by itsel. ie: x y + 4 = 0 dy I y = x then = 1 I ( x) y = then. I ( x) = y then Eg.1: Given the equation or a circle with radius 1, ind the slope o the tangent to the circle at ( 5,1). x + y = 169 Eg.: Dierentiate x b) ( x + y) ( ) xy + 8π = 1 4 a) y + 6xy = 10 UL9: Unit Review Day # 1 Homework: p. 564 #, bc, 4, 5a, 6, 7, 9ab p.110 #abc, 5bcde, 6ab, 7c, 11a, 1, 1, 18, abcd, abc, 8cdg, 9, 0abcd p.564 #ad 5b UL10: Unit Review Day # UL11: Unit # Test Page 11 o 11