Calculus: Analytic Geometry and Calculus, with Vectors

3.6 The chain rule and differentiation of elementary functions 175

26 The preceding problem involved three functions and the Newton notation
for derivatives. This problem requires use of the Leibniz notation. Supposing
that "y is a function of x and x is a function of t so y is a function of t," and that
each function has three or more derivatives, write formulas for the first three
derivatives of y with respect to t. Finally, check your answers against those of
the preceding two problems. Partial ans.:

dydydx

(1) dt dx dt

d2y dy d2x d23, (dxl2

(2) dt2 dx dt2 + dx2 \dt /

d3y dyd3x d23, d2x dx d3y dx 3

(3) dt3 = dx dt3 + 3dx2dt2 dt + Ti % dt

27 Suppose we momentarily agree that the first of the formulas

dy dy dx d2y d2y (dxl2

dt = dx dt dt2 = dx \dt I (?)

is true "because" we get a correct result by canceling dx's from the right side.
Show that we should not apply the same "reasoning" to the second formula.
28 Read Theorem 3.65 and observe that the hypotheses are satisfied if
f(x) = 1 + x + x2, g(x) = 0, and u = g(x) = 0 for each x, while y = f(u) for
each u so that y = f(g(x)) = 1 for each x. Hence the conclusion of the theorem
implies that
dy dy du

dx=dudx

for each x. Observe that dy/dx = 0 and du/dx = 0 for each x. Our major
question now appears. Is there a reason for uneasiness about the meaning of
dy/du when u = 0 for each x? Remark and ans.: This question was raised by an
extremely sane person who happened at the moment to be thinking too much
about the manner in which we read dy/du and too little about the meaning of
dy/du. According to our basic definition, dy/du is f'(u), the derivative of f at
u. Since f(x) = 1 + x + x2 for each x, we find that f' (x) = 1 + 2x for each x,
so f'(u) = 1 + 2u for each is. Thus, dy/du = 1 + 2u. If it happens that
u = 0 for each x, then dy/du = 1 for each x. We have no reason to be uneasy
unless we manufacture trouble by recreating old tales about varying variables
that we sometimes call galloping numbers. The notation of Leibniz is often
more convenient than that of Newton, but it is also more likely to engender
mental aberrations. Nobody expects u to be galloping around while we calculate
f' (u)
29 Is the function f for which f(x) = JxI an elementary function? Remark
and ans.: This is a tricky question. An intelligent perso d make an incor-
rect guess until he discovers or is reminded that jxI _ 2 The function f is
elementary, but f'(0) does not exist.
30 Let p and q be positive integers. Let y(O) = 0 and let

y(x) = xD sin z (x 0 0).