1 / 8

6.2 Rates of Change for Composite Functions - The Chain Rule

6.2 Rates of Change for Composite Functions - The Chain Rule. Given: y = (4 – 3 x ) 3. then y = u 3. let u = 4 – 3 x. y is a function of u and u is a function of x is a composite function.

maxine
Download Presentation

6.2 Rates of Change for Composite Functions - The Chain Rule

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 6.2 Rates of Change for Composite Functions - The Chain Rule Given: y = (4 – 3x)3 then y = u3 let u = 4 – 3x y is a function of u and u is a function of x is a composite function. Therefore if h(x) = (fog)(x), then h´(x) = f ´(g(x))·g´(x)

  2. Proof of the Chain Rule multiply by Du/Du provided Du ¹0 if both limits exist (def. of derivative)

  3. Example 1: Determine if y = (4x + 6)3 = 3(4x + 6)2(4) = 12(4x + 6)2 If we had expanded y = (4x + 6)3 and found the derivative of the product, the result would have been the same.

  4. The Chain Rule with the Power Rule = n[g(x)]n–1·g´(x), where n is a constant Take the derivative of the ‘outer’ function multiplied by the derivative of the ‘inner’ function. Example 2: Determine f ´(s) if f(s) = (2s3 – 5)4 4(2s3 – 5)3 (6s2) f ´(s) = derivative of outer · derivative of inner f ´(s) = 24s2(2s3 – 5)3

  5. Example 3: Determine

  6. Differentiating a quotient with constant numerator. Example 4: Determine f ´(t) f(t) = 5(4t + 3)–3 f ´(t) = (–3)(5)(4t + 3)–3–1 (4) derivative of outer · derivative of inner f ´(t) = (–60)(4t + 3)–4

  7. Example 5: Determine y´ This is the composition of y = u3 and

  8. Example 6: A cylindrical barrel has a radius of 1.2 m. If water is pouring in at a rate of 3 m3/min, at what rate is the height of water in the barrel increasing? 1.2 m V = pr2h Since r = 1.2 m, Using the chain rule we have: The height of water is changing at a rate of 0.66 m/min.

More Related