Chain Rule. Let M, N and Q be differentiable manifolds, and let φ : M ?> N and N ?> Q be differentiable mappings. Prove that
....
or simply written
.....
[Comment: To familiarize yourself with notations in Differential Geometry try to check the form that the above equality takes when you express the (differentials of th

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What can you say about the differentiability of fg at x = c in each of the following cases? (Here the function fg is defined by fg(x) = f(x)g(x)).
(a) f is differentiable at c, but g is not.
(b) f is not differentiable at c, and neither is g.
(c) both f and g are differentiab

Suppose fâ?¶Râ?'R is twice differentiable with both f' and f'' continuous in an interval around 0. Suppose further that f(0)=0. Let
h(x)={f(x)/x, if xâ? 0,
f^' (0), if x=0.
Show that
(a) h is differentiable at x=0.
(b) h is differentiable at x=0 with h^' (0)=1/2 f^'' (0).
(c) h' is continuous at x=

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Let a sequence xn be defined inductively by . Suppose that as and . Show that
.
(Note that " " refers to "little oh")
HINT: Use the Mean-Value Theorem and assume that F is a continuously differentiable function.

Let g:[0,1]->R be twice-differentiable (i.e both g and g' are differentiable functions) with g''(x)>0 for all x belong to [0,1].if g(0)>0 and g(1)=1 show that g(d)=d for some point d belong to (0,1) if and only if g'(1)>1.

Let
f(x)={(x^3)cos(1/x) if xâ? 0, 0 if x=0,
and
g(x)={(1/x)sin(x) if xâ? 0, 0 if x=0.
a) Using the definition of the derivative show that f is differentiable at 0 and determine f '(0).
b) Is g differentiable at 0? Justify your answer.
c) Show that f ' ^(x) and g ' ^(x) exist for xâ? 0 and determine their value

Work the following problem using an Excel. Annotate and highlight the spreadsheet with the answers required below.
Bill's Manifolds, Inc. produces made to order special manifolds for motorcycles. The total time from receipt of order until the customer receives the product consists of two phases: 1) manufacturing time and 2

Prove that if f is differentiable at x, then
(see attachment)
Also, show that for some functions that are NOT differentiable at x, this limit still exists.
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Let f: I →ℜ where I is an open interval containing the point c, and let k ∈ ℜ. Prove the following
1. f is differentiable at c with f ′(x) = k iff lim h→0 [f(c+h) - f(c)]/h=k
2. If f is differentiable at c with f ′(c) =