13. If x1+ 2ax3y — ay3 = 0, shew that dy when x=0 and y = 0. = 0, or ±√2, dx 14. If x2- ay3+2axy2+3ax3y=0, shew that or 3, when x=0 and y = 0. y=0. 15. If ax3+x3y — ay3 = 0, shew that dy — : = dx 16. If x2y2 = (a2 — y3) (b+ y)2, shew that when x= 0 and y=-b. 3 dy dx 2 (y2+ x2 - 2x)2, 17. If (y2 — x2) (x − 1) ( x − 2) = find Results dy dx when x and y vanish, and when x=1, y=1. 16 u2 + x2 + y2+ z2 = c2, 7. Given y3 + x3 — 3axy = 0, shew that dy find 9. If y=(x, y, u) and f (x, y, u) = 0, find du 11. If u = a2 + √(sec xy), find da (1) when x and y are dx' independent, (2) when x+y=a. 12. If a+√(sec xy) = 0, find dy dx dy y√(sec xy) tan xy + 2a yx2-1log a Result = dx x√(sec xy) tan xy+2a**x log a logx dy 13. If x+2ax3y - ay3 = 0, shew that = 0, or ±√√2, dx when x= 0 and y = 0. 14. If x*— ay3+2axy®+ 3axy=0, shew that dy 16. If x3y2 = (a2 — y2) (b+ y)2, shew that when x=0 and y =—b. dx dy b = + = = 2 (y2+ x2 — 2x)3, 18. find dy when a and y vanish, and when x=1, y=1. dx If y* — yo + 3xy — 2x2 = 0, find dy when x = 0. dx u3 + x2 + y2+z2 = c2, CHAPTER XII. CHANGE OF THE INDEPENDENT VARIABLE. 196. In Art. 60 we have proved the equation and we now proceed to some extensions of these formulæ. Given x and y, both functions of a third variable z, it is required to express the successive differential coefficients of y with respect to x, in terms of those of y and x with respect to z. We have dy_dy dz = dx dz dx by (2), dz by (1). dx |