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B2.2 & B2.3 - Product Rule and Quotient Rules

B2.2 & B2.3 - Product Rule and Quotient Rules. MCB4U & IB Math HL/SL - Santowski. (A) Review. The derivative of a sum/difference is simply the sum/difference of the derivatives i.e. (f + g)` = f ` + g`

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B2.2 & B2.3 - Product Rule and Quotient Rules

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  1. B2.2 & B2.3 - Product Rule and Quotient Rules MCB4U & IB Math HL/SL - Santowski

  2. (A) Review • The derivative of a sum/difference is simply the sum/difference of the derivatives i.e. (f + g)` = f ` + g` • The power rule tells us how to find the derivative of any power function y = xn which works for any real value of n

  3. (B) Product Rule – An Investigation • Now the question concerns products of functions  is the derivative of a product of functions the same as the product of the derivatives? i.e. is (fg)` = f` x g` ?? • Let's investigate with a product function h(x) = f(x)g(x) where f(x) = x² and g(x) = x² - 2x. • Thus h(x) = x²(x² - 2x)

  4. (B) Product Rule – An Investigation • Trial 1  if we go with our idea that (fg)` = f` x g`  then (fg)` = (2x)(2x - 2) = 4x² - 4x • We can graph h(x) on the GC, graph the derivative and then program in 4x² - 4x and compare it to the calculated derivative from the GC. We will find that the two do not match up!!! • If we simply tried expanding h(x) = x4 - 2x3 and then taking the derivative, we would get 4x3 – 6x2. If we program 4x3 – 6x2 into the GC, we find that we match the GC generated derivative exactly.

  5. (B) Product Rule – An Investigation

  6. (C) Power Rule - Derivation • So why does (fg)` = f ` x g` not work? • Let’s go back to limits and basic principles to find what the differentiation technique should be if we wish to find a derivative of a product • Let K(x) = [f(x)] x [g(x)] • Then K `(x) = lim h0 1/h[K(x + h) – K(x)] • And K `(x) = lim h0 1/h [ f(x+h)g(x+h) – f(x)g(x)] which gets simplified as on the next slide:

  7. (C) Power Rule - Derivation

  8. (D) Power Rule - Examples • ex 1. Find the derivative of f(x) = 3x4(5x3 + 5x - 7) • ex 2. Find the derivative of f(x) = (x4-4x3–2x2+5x+2)2 • ex 3. Find the equation of the tangent to the function f(x) = (2x + 4)(3x3 – 3x2 + x - 2) at (1,-6)

  9. (E) Derivatives of Rational Functions – The Quotient Rule • Since the derivative of a product does not equal the product of the derivatives, what about a quotient? • Would the derivative of a quotient equal the quotient of the derivatives? • Since quotients are in one sense nothing more than products of a function and a reciprocal  we would guess that the derivative of a quotient is not equal to the quotient of the derivatives

  10. (F) Quotient Rule - Derivation • First, set up a division and then rearrange the division to produce a multiplication so that we can apply the product rule developed earlier

  11. (G) Examples Using the Quotient Law • Differentiate each of the following rational functions • ex 1. • ex 2. • ex 3.

  12. (H) Internet Links • Calculus I (Math 2413) - Derivatives - Product and Quotient Rule • Visual Calculus - Calculus@UTK 3.2 • solving derivatives step-by-step from Calc101

  13. (I) Homework • IB Math HL/SL  Stewart, 1989, Chap 2.4, Q2eol,3eol,5,6,9b as well as Chap 2.5, Q2eol,3eol,4-7 • MCB4U  Nelson text, Chap 4.4, p301 Q5,7,9,10,13 and Chap 5.4, Q2,4,7,8,15,17

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