6.2 Rules for Differentiation 315- Prove the general power rule for real number exponents: If f is differen-
tiable then v c E JR, d~ [f(x)r = c [f(xW-
1
f'(x). - Prove that "ix-:/:-0, d~ ln lxl = ~-
15. Use the "logarithmic differentiation" technique learned in elementary cal-
culus to prove that if f and g are differentiable and f is positive, then
ix [f(x)]^9 (x) = [f(x)]^9 (x) (g'(x) ln f(x) + g(x) ~(~?).16. Assume formula (a) of Table 6.1. Use this, trigonometric identities, the
algebra of derivatives, and the chain rule if necessary, to derive formulas
(b )-(f).- In Example 6.1.10 we proved that the function. f(x) = { x x.
(^2) sin l if x f:. 0 }
0 if x = 0
is differentiable at 0.
(a) Prove that f is differentiable everywhere on R
(b) Prove that f' is continuous everywhere except at 0. [To prove that f'
is not continuous at 0, you may find the sequential criterion helpful.]
- Use mathematical induction to prove Leibniz's rule: "in EN, if f and g
are n times differentiable at x, then (fg)(n)(x) = k~O (~) J(k)(x)g(n-k)(x),
where J(k) and g(k) denote the kth derivatives of f and g, J<^0 l = f,g(o) = g, and (~) = k!(nn~ k)!. [See Exercise 1.3.23.]- A function f is said to be periodic with period p > 0 if "ix E JR,
f(x+p) = f(x). [See Exercise 5.4.22.] Prove that if f: JR---+ JR is periodic
with period p and differentiable on some interval [a, a+ p), then f is
differentiable everywhere on JR and f' is periodic with period p.
*20. Periodic Extensions: Suppose f: [a, b) ---+ R Define f: JR---+ JR by
f(x) = f ( x - (b - a) l ~ =: J), where lxJ denotes the "greatest integer
function."^6 Prove that(a) "ix E JR, x - (b - a) l ~ =: J E [a, b).
(b) f is periodic, with period b - a, and J1ra,b)
periodic extension of f to R)- The greatest integer function lxJ is defined in Example 5.2.16.
f. (We call f the