b) If f is differentiable at a and g is differentiable at f(a), then (gof)" (a) = g'(f (a)) f" (a) +g" (f(a)) (f'(a))². c) If the nth-order derivatives f(n) (a) and g(n)(a) exist, then (f+g)) (a) = f(n)(a) + g(n) (a). d) If the nth-order derivatives f(n)(a) and g(n)(a) exist and are nonzero, then (9)") (a) = g(a) f(n)(a) + (-1)" f(a)g(") (a) gn+¹(a)
b) If f is differentiable at a and g is differentiable at f(a), then (gof)" (a) = g'(f (a)) f" (a) +g" (f(a)) (f'(a))². c) If the nth-order derivatives f(n) (a) and g(n)(a) exist, then (f+g)) (a) = f(n)(a) + g(n) (a). d) If the nth-order derivatives f(n)(a) and g(n)(a) exist and are nonzero, then (9)") (a) = g(a) f(n)(a) + (-1)" f(a)g(") (a) gn+¹(a)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Plz solution to a b and d
Transcribed Image Text:b) If f is differentiable at a and g is differentiable at f(a), then
(gof)" (a) = g'(f (a)) f"(a) +g" (f (a)) (f'(a))².
c) If the nth-order derivatives f(n) (a) and g(n) (a) exist, then
(f+g)) (a) = f(n) (a) + g(n) (a).
derivatives f(n) (a) and g(n)(a) exist and are
d) If the nth-order
nonzero, then
(9)
(n)
(a) =
g(a) f(n) (a) + (-1)" f(a)g(") (a)
gn+¹(a)
Transcribed Image Text:4.2.0. Suppose that I is an open interval containing a, and that f. g. h: I→ R.
Decide which of the following statements are true and which are false.
Prove the true ones and provide counterexamples for the false ones.
a) If f, g, and h are differentiable at a, then
(fgh)'(a) = f'(a)g(a)h(a) + f(a)g'(a)h(a) + f(a)g(a)h' (a).
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