§2.3 The Chain Rule and Higher Order Derivatives. The student will learn about. composite functions,. the chain rule, and. nondifferentiable functions. Composite Functions. Definition. A function m is a composite of functions f and g if - PowerPoint PPT Presentation
Text of §2.3 The Chain Rule and Higher Order Derivatives
*2.3 The Chain Rule and Higher Order DerivativesThe student will learn aboutcomposite functions, the chain rule, andnondifferentiable functions.
*Composite Functions Definition. A function m is a composite of functions f and g if m (x) = f g = f [ g (x)]This means that x is substituted into g first. The result of that substitution is then substituted into the function f for your final answer.
*ExamplesLet f (u) = u 3 , g (x) = 2x + 5, and m (v) = v. Find: f [ g (x)] = g [ f (x)] =g (x3) = m [ g (x)] = f (2x + 5) =(2x + 5)3m (2x + 5) = 2x 3 + 5 2x + 5
*Chain Rule: Power Rule. We have already made extensive use of the power rule with xn, We wish to generalize this rule to cover [u (x)]n, where u (x) is a composite function. That is it is fairly complicated. It is not just x.
*Chain Rule: Power Rule. That is, we already know how to find the derivative of f (x) = x 5We now want to find the derivative of f (x) = (3x 2 + 2x + 1) 5What do you think that might be?
* General Power Rule. [Chain Rule]If u (x) is a function, n is any real number, andIf f (x) = [u (x)]nthenf (x) = n un 1 uor* * * * * VERY IMPORTANT * * * * *Chain Rule: Power Rule.
*ExampleFind the derivative of y = (x3 + 2) 5.Let the ugly function be u (x) = x3 + 2. Then5(x3 + 2)3x24= 15x2(x3 + 2)4
*ExampleFind the derivative of y =Rewrite as y = (x 3 + 3) 1/2 = 3/2 x2 (x3 + 3) 1/2Then y = 1/2Then y = 1/2 (x 3 + 3) 1/2Then y = 1/2 (x 3 + 3) 1/2 (3x2)Try y = (3x 2 - 7) - 3/2 y = (- 3/2) (3x 2 - 7) - 5/2 (6x)= (- 9x) (3x 2 - 7) - 5/2
*ExampleFind f (x) if f (x) = We will use a combination of the quotient rule and the chain rule.Let the top be t (x) = x4, then t (x) =4x3Let the bottom be b (x) = (3x 8)2, then using the chain rule b (x) = 2 (3x 8) 3 =6 (3x 8)
*Remember Def: The instantaneous rate of change for a function, y = f (x), at x = a is: This is the derivative.Sometimes this limit does not exist. When that occurs the function is said to be nondifferentiable.
*Remember Def: The instantaneous rate of change for a function, y = f (x), at x = a is: This is the derivative and a graphing way to represent the derivative is as the slope of the curve. This means that at some points on some curves the slope is not defined.
*"If a function f "
*Summary. Ify = f (x) = [u (x)]nthenNondifferentiable functions.