MATH 234, Calculus for BusinessLecture 10, Textbook Sections 4.3The Chain Rule for DerivativesFebruary 22nd, 2017Announcements HW 4 is due Thursday, 2/23, 11:59pm; late HWs are not accepted.Example 1. Compute the derivative of f(x) = (2x10+ 4)(x2+ 1).Name the rules that were applied at each step.Solution.Question 1. Is it possible to differentiate√2x + 5 or√3x + 4?Remark. Yes. Use the limit definition of the derivative (see Exam 1).However, the power rule, product rule, quotient rule, etc will not work.To make a more efficient method, first find the obstacle.Example 2. Let f(x) =√x and let g(x) = 3x + 4. Compute their derivatives.Solution.Remark. The function√3x + 4 is a composite function.Using the functions above,√3x + 4 = f(3x + 4) = f (g(x)).The obstacle is to find a way to differentiate composite functions.1Definition. The composition of f(x) with g(x) is .The composition of g(x) with f(x) is .Example 3. Let f(x) =√x and g(x) = 3x + 4. Thenx f(x) g(x) f(g(x)) =√3x + 4 g(f (x)) = 3√x + 40 0 41 1 74 2 169 3 31WARNING!. In general, f(g(x)) 6= g(f (x)). For compositions, the order matters.Theorem 1 (The Chain Rule). Let f(x) and g(x) be differentiable functions, thenf(g(x))0=Mnemonic: “Diff out comp in, mult diff in”Example 4. Compute the derivative of√3x + 4Solution.Example 5. Compute the derivative of (x2+ 10)500Solution.2Example 6. An oil pipeline ruptures somewhere in the Midwest.Aerial pictures show that the pollution expands in a circular manner.Engineers estimate that, after t hours, the radius of the disk (in miles) isr(t) = 50 ∗ t1/2+ 90 ∗ t1/3How fast is the polluted region growing, in square miles per hour?Solution.Example 7. A sample list of functions and their derivatives isFunction Derivative of Function(2x3+ 3x + 3)1/616∗ (2x3+ 3x + 3)−5/6∗ (6x2+ 3)px2+√x12px2+√x∗2x +12√x(x2+ x−5)10+√4x + 7 10(x2+ x−5)9∗ (2x − 5x−6) +2√4x + 7f(g(h(x)) f0(g(h(x)) ∗ g0(h(x)) ∗ h0(x)q3 +p7 +√x12q3 +p7x +√x∗12p7 +√x∗12√xFill in the work, and try to get the same answer. For the last two examples, use the chain rule twice.Theorem 2 (The Chain Rule, with an alternate notation). Let y be a function of u: y = f(u). Let u be afunction of x: u = g(x). Then y is a function of x: y = f (u) = f(g(x)), anddydx=Remark. Do not “cancel” dy, du or dx, etc. They are meaningless expressions.Example 8. Differentiate y = (x2+ 10)500using the alternate notation form of the chain rule.Solution.3Example 9. Differentiate y =√3x + 4 using the alternate notation form of the chain rule.Solution.Example 10. Let y =√u + u5and u = x3+ 2x. Computedydx.Solution.Example 11. Differentiate(5x2+ 3)1/3x2+ 2Solution.4Example 12. Differentiatep7x +√x3+ 4Solution.A (more challenging) list of functions with their derivatives isFunction Derivative of Functionrx − 1x + 11(x + 1)√x2− 1(x2+ 3)4(2x3− 5)32x(x2+ 3)3(2x3− 5)2(17x3+ 27x − 20)x3− 12x3+ 1436x2(x3− 1)3(2x3+ 1)5x√1 − 4x21(1 − 4x2)3/2(x − 1)√x2− 2x + 22x2− 4x + 3√x2− 2x + 2This list comes from “Schaum’s Outlines: Calculus, 4th ed.”Fill in the work for yourself, and see if you can get the same
View Full Document
Unlocking...