9.43
THE CHAIN RULE
9.44
9.45
Assume that F and G are differentiable functions such that F'(x) = -G(x) and G'(x) = -F(x). Let
H(x) = (F(x)]2-[G(x)]2. FindH'M.
H'(x) = 2F(x) • D x F(x) - 2G(x)-D x G(x) = 2F(x)[-G(x)] - 2G(x)[-F(x)] = ~2F(x)G(x) + 2F(x)G(x) = 0.
9.46
Assume e>0. Choose ^ >0 such that \g(u) ~ g(f(a))\ < e whenever \u-f(a)\<8 l . Then choose
5 >0 such that \f(x) - f(a)\ < 5, whenever \x - a\ < S. Hence, if \x - a\ < S, \g(f(x)) - g(/(a))| < e.
9.47
Show that
When x>0, D,\x\ = D x (x) = I = \x\/x. When x<0, D x \x\ = D x (-x) = -1 = -xlx = \x\lx.
9.48
Find a formula for D x \x
2 + 2x\ (x * 0, -2).
By the chain rule and Problem 9.47,
9.49
Give a justification of the rule
61
Let u=f(v) be a one-one, differentiable function. Then the inverse function v = g(u) = g(f(v)) is
differentiable, and the chain rule gives
Writing dvidu and duldv for g'(u) and/'(i>) in (2), we get (1).
for
If /is continuous at a and g is continuous at /(a), prove that g°f is continuous at a.