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Calc 2.3a


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Calc 2.3a

  1. 1. 2.3A PRODUCT AND QUOTIENT RULE Goals: To differentiate a product To differentiate a quotient To find derivative of tangent, cotangent, secant and cosecant To find higher-order derivatives
  2. 2. <ul><li>In other words, </li></ul><ul><li>first•derivative of last + last •derivative of first </li></ul><ul><li>With a southern accent, </li></ul><ul><li> First de-last + last de-first </li></ul>This rule can be extended to three products or more, see page 119 at the bottom of the page!
  3. 3. EXAMPLE 1, P. 120 USING THE PRODUCT RULE <ul><li>Derivative of a product CANNOT, in general, be written as product of two derivatives. </li></ul><ul><li>Find the derivative of h(x) = (2x – 3x 2 )(4x – 5) </li></ul><ul><li>Using rule: </li></ul><ul><li>h’(x) = (2x – 3x 2 )(4) + (4x – 5)(2 – 6x) </li></ul><ul><li> = 8x – 12x 2 + 8x – 24x 2 – 10 + 30x </li></ul><ul><li> = -36x 2 + 46x – 10 </li></ul><ul><li>Not using rule but foiling first: </li></ul><ul><li>h(x) = -12x 3 + 23x 2 – 10x which derives to same thing. </li></ul>
  4. 4. <ul><li>In many cases you MUST use product rule! </li></ul><ul><li>Ex 2 p. 120 </li></ul><ul><li>Find the derivative of y = 4x 3 sin x </li></ul>Putting in factored form as much as possible allows us to find the places where a horizontal tangent occurs more easily (where y’ = 0)
  5. 5. <ul><li>Ex 3 p. 120 </li></ul><ul><li>Find the derivative of y = 3xsinx – 4cosx </li></ul><ul><li>(identify the products within the function!) </li></ul><ul><li>Notice the underlined part is applying the product rule </li></ul>Use the product rule if __________________________. Use Constant rule if ____________________________. both factors are variable one of the factors is constant.
  6. 6. Break out that cowboy accent! Low dehigh – high delow all over low squared!
  7. 7. <ul><li>Ex 4, p. 121 Using the quotient rule </li></ul><ul><li>Find the derivative of </li></ul>In general, see if numerator factors, and keep denominator in a factored form Note liberal use of parentheses!
  8. 8. <ul><li>Warning – pay special attention to the subtraction required in the numerator – more points lost there than anywhere! </li></ul><ul><li>Sometimes a rewrite is needed here too: </li></ul><ul><li>Ex. 5 p. 122 Rewriting before differentiating </li></ul><ul><li>Find an equation of the tangent line to f(x) at (-1, 1) </li></ul><ul><li>So </li></ul><ul><li>and tangent line is y = 1 </li></ul>
  9. 10. CAUTION: NOT EVERY QUOTIENT DESERVES THE QUOTIENT RULE!!!! <ul><li>Original Function Rewrite Derive Simplify </li></ul>
  10. 11. ASSIGNMENT 2.3A <ul><li>p. 126 #2-12 ev, 13-41 eoo, 53,65,69 </li></ul>