Chain Rule
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Chain Rule

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Chain Rule Chain Rule Presentation Transcript

  • THE CHAIN RULE Chapter 3 Section 6
  • The Purpose of Chain Rule
    • The chain rule is an effective yet easier way of finding the derivative of more complicated equations.
    • Look at this problem: y = (x 3 + 5x) 7
  • Chain Rule Contd.
    • y = (x 3 + 5x) 7
    • Formula to find derivative of above problem:
    • y = (u) 7 ; u= x 3 + 5x : get derivative of “u”, which is:
    • 3x 2 + 5 = du : now, to find y’,
    • y’= 7u 7-1 x du or 7u 6 x du : plug in “u” and “du” to get:
    • 7(x 3 + 5x) 6 (3x 2 + 5) : distribute the 7 and multiply
    • y’= (21x 2 + 35)(x 3 + 5x)
  • The Chain Rule & Trig. Functions
    • Find the derivative using chain rule:
    • y = sin 3 2x find y’
    • 1: sin 3 2x = (sin 2x) 3 [changing makes it easier]
    • our “u” is: (sin 2x)
    • our “du” is: 2cos 2x
    • y= u 3 x du
    • y’= 3u 2 x du ==> plug in “u” and “du” into equation.
    • y’= 3(sin 2x) 2 (2 x cos 2x)
    • y’= 6(sin 2x) 2 (cos 2x)
  • Using Inside-Outside Method
    • The following problem can be solved 2 ways:
    • y= sin(3x + 1)
    • -The first way we’ll do is the chain rule:
    • “ u” = 3x + 1 making our “du” = 3
    • y= sin u x du y’= cos u x du ==> plug in “u” and “du”
    • y’= cos (3x +1) x 3 = 3 cos (3x + 1)
  • Using Inside-Outside Method contd.
    • y= sin(3x + 1)
    • To solve using Inside-Outside Method:
    • Take the derivative of the outside of the equation (sin(3x +1))
    • Take the derivative of the inside of the equation (3x +1)
    • Derivative of outside (sin (3x +1)) = cos(3x +1)
    • Derivative of inside (3x +1) = 3
    • Put them together to GET:
    • y’= 3 cos(3x + 1)
  • Chain rule and Radicals Solve for y’ using the Chain rule To SOLVE: = (9x 2 +4) 1/2 “u” = 9x 2 +4 “du” = 18x y = u 1/2 x du y’= 1/2u -1/2 x du [plug in “u” and “du”] y’= 1/2(9x 2 + 4) -1/2 x 18x : multiply 18x by 1/2 to get: 9x (9x 2 + 4) -1/2 =
  • Chain rule and Radicals Find y’ To solve, you will use the chain rule and inside outside rule. = (tan(3x)) 1/2 u= tan (3x) To find du, use inside outside rule. So, u = tan (3x) Derivative of tan (3x) [outside: sec 2 (3x)] Derivative of (3x) [inside: 3] du = 3 tan(3x) continue on next slide…
  • Chain rule and Radicals
    • u= tan (3x) and du= 3 tan (3x)
    • y= u 1/2 du | y’=1/2u -1/2 x du [plug in “u” and “du”]
    • y’= 1/2(tan 3x) -1/2 x 3 tan 3x
    • =
  • THE END!