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Math 142 Lecture Notes for Section 4.3-4.4 Section 4.3 - 1 The Chain Rule Definition 4.3.1: If g is a differentiable at x and f is differentiable at g(x), the the composite function h(x) = f (g(x)) is differentiable at x and it’s derivative is In Leibniz notation if y = f (u) and u = g(x) are both differentiable functions, then Example 4.3.2: Use the chain rule to establish some general derivative rules: n If y = [f (x)] , then If y = ef (x) , then If y = bf (x) , then If y = ln [f (x)], then If y = logb [f (x)], then Math 142 Lecture Notes for Section 4.3-4.4 Example 4.3.3: Differentiate the following: (a) f (x) = (3x2 + 4x)3 (b) f (x) = (c) y = (t2 √ 2x2 − 4 −2 + 2t + 1)3 (d) f (x) = ln(3x ) 2 Math 142 Lecture Notes for Section 4.3-4.4 (e) f (x) = 8x4 ex 3 √ (f) f (x) = 3( x − ex )2 (g) f (x) = log3 (x2 + 4) (h) f (x) = ln( x+3 ) x−2 3 Math 142 Lecture Notes for Section 4.3-4.4 4 3 3 ln x2 − x4 ) (i) f (x) = x2 Example 4.3.4: Find the value(s) of x where the tangent line is horizontal for x3 (5x − 4)2 Example 4.3.5: Given the following information g(2) = 6, g 0 (2) = 3, f 0 (6) = 5, f 0 (3) = 1, f 0 (2) = 7. If h(x) = f (g(x)), then what is h0 (2)? Math 142 Lecture Notes for Section 4.3-4.4 5 Example 4.3.6: Let y = ln u and u = 3x2 + 5 dy . Use the second form of the derivative to find . x dx Example 4.3.7: If James invests 10,000 into a money market account offered by Skyfall Ltd. with an annual interest rate of 7.5% compounded continuously, how fast is the balance growing after 25 years? Suggested Homework: (4.3) 1-37(odd), 41,45. (4.4) 1-73(odd).