253/826 It is common to present the formula in (7.3.2) in the deceptively simple way: dx dy dy/dx (7.3.4) 234 CHAPTER 7 / DERIVATIVES IN USE as if dx and dy could be manipulated like ordinary numbers. Formula (7.3.4) shows that similar use of the differential notation for second derivatives fails drastically. The formula "d2x/dy2 = 1/(dy2/d2x)", for instance, makes no sense at all. EXAMPLE 7.3.4 Suppose that, instead of the linear demand function of Example 4.5.4, one has the log-linear function In Q = a - b In P. (a) Express Q as a function of P, and show that de/dP = -be/P. (b) Express P as a function of Q, and find dP/de. (c) Check that your answer satisfies the version dP/de = 1/(de/dP) of (7.3.4). Solution: (a) Taking exponentials gives Q = ea-binP = ed(elp)-b = eap-b, from which it follows that de/dP = -beap-b-1 = -bQ/P. (b) Solving Q = eap-b for P gives P = ea/bQ-1/b, so dP/do= (-1/b)ea/bQ-1-1/b. (c) From part (b) one has dP/do = (-1/b)P/Q = 1/(de/dp).(SM 5. Use (7.3.4) to find dx/dy when: (a) y = ex-5 (b) y = In(e-*+ 3) ( c) xy' - xy = 2x