(9/30/08)Math 10A. Lecture Examples.Section 3.6. The Chain Rule and inverse functions†Example 1 What is the derivative of y = 3 ln x − 4x3Answer:dydx=3x− 12x2Answer: F0(1) =54Example 3 Find an equation of the tang ent line to y = ln x at x = 2 and draw it withthe curve.Answer: Tangent line y = ln(2) +12(x − 2) • Figure A3x1 2 3 4y2−11y = ln xFigure A3Example 4 What is the derivative of Z(x) = arcsin x + arctan x at x = 0?Answer: Z0(0) = 2Example 5 Find the derivative of y = x2arctan x at x = 3.Answer: y0(3) =910+ 6 arctan(3)Example 6 What is Y0(x) for Y(x) = (arcsin x)2?Answer: Y0(x)) =2 arcsin xp1 − x2†Lecture notes to accompany Section 3.6 of Ca lculus by Hughes-Hallett et al.1Math 10A. Lecture Examples. (9/30/08) Section 3.6, p. 2Example 7 (a) Give a formula for the angle ψ in Figure 1 in terms of the length yof the opposite side o f the triangle. (b) At what rate is ψ increasing at amoment when y = 2 meters and y is increasing12meter per minute?1 meteryψFIGURE 1Answer: (a) ψ = arctan y (b)dψdt=110meters per minuteInteractive ExamplesWork t h e following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡Section 3.1: Examples 1, 2, and 3Section 3.6: Examples 4 and 6‡The chapter and section numb ers on Shenk’s web site refer to his calculus manuscript and not to the chapters and sectionsof the textbook for the
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