314 Chapter 6 11 Differentiable Functions
- Prove that if f is differentiable at xo and f (xo) =f. 0, then
_!}__ [-1 ] -f'(x)
dx f(x) - J2(x) · - Suppose f : JR-> JR is differentiable at x 0 , and define g : JR-> JR by
the given formula. Use the theorems of this section to prove that g is
differentiable at xo and find g'(xo) in terms of xo, f(-), and!'(-).
(a) g(x) = x^3 f(x^2 )
(c) g(x) = [f(x^3 ) ]
5(b) g(x) = x^4 j3(x)
f ( l)
(d) g(x) = --f-, if x =f. 0
x- We define a function f JR -> JR to be an odd function if 'Vx E JR,
f(-x) = -f(x), and an even function if 'Vx E JR, f(-x) = j(x). Sup-
pose a function f : JR -> JR is differentiable everywhere. Prove:
(a) If f is an odd function, then f' is an even function;
(b) If f is an even function, then f' is an odd function. - State and prove a product rule for hgf.
- State and prove a chain rule for hog of.
- Suppose f is differentiable at an interior point x 0 of its domain, and g is
differentiable at f(xo), an interior point of its domain. Find the flaw in
the following "proof" of the chain rule:
1. Im -'-~'-'--'--------'-~-'--'----'-(go J)(x) - (go f)(xo)
x-> xo X -Xo
= lim (g(f(x)) -g(f(xo)). f(x) - f(xo))
x->xo f(x) - f(xo) x - Xo
1. g(f(x)) - g(f(xo))
1
. f(x) - f(xo)
Im · Im ~~~~-
f ( x )-> f ( xo) f(x) - f(xo) x->xo x - Xo
(using Theorem 4.2.23).
= g'(f(xo) · f'(xo).
Then show that the proof is valid if f is strictly monotone in a neighbor-
hood of xa.
9. Determine where the function Jx + Jx +ft is differentiable, and find
its derivative.
Prove Corollary 6.2.8. [Hint: x~ = ( x*) m .J
Prove Corollary 6.2.10. [Hint: Use Exercise 5.6.16 to convert to base e.]
Prove Corollary 6.2.11. [See hint for Exercise 11.]