Differentiation

#15-oct

Definition.

is said to be differentiable at if
then we write

#16-oct

let and is not a single ton

Definition.

Let be a function suppose then is said to be differentiable at if exist then

Theorem.

Suppose is differentiable at then is continuous at converse is not true

Proof.

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Example

Consider
take

Theorem.

suppose are functions differentiable at then

is differentiable at for and
is differentiable at and

Proof.

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Theorem.

Suppose are functions diffrentiatable ar
if then is same interval I containing on I is well defined and diffrentiable at
with

Proof.
As is differentiable at it is continuous at

takr $ε$ as we have $ε$
so

ε

so for

Take

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Corollary.

Suppose is a real polynomial

then at any is differentiable at and

Definition.

Let be a function

said to be differentiable it is differentiable at every
is a function on I called the first Derivative of f
if is differentiable at every then is said to be twice differentiable and is a function on I defined by
for is differentiable at every then is said to be times differentiable
is said to be infinite differentiable if is n-times differentiable for every
is said to be continuously differentiable if is differentiable and is continuous

Theorem ( Carathéodory's theorem).

Let be a function. Then is differentiable at iff there exist a function continuous at and satisfying

In such a case is unique and

Proof.
suppose is differentiable at
take

Then clearly satisfies * and is continuous at

If there exist satisfies and continuous then so is differentiable and
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Theorem (chain rule).

suppose and are function such that take suppose is differentiable at and is differentiable at then is differentiable at and

Proof.

by Carathéodory theorem Carathéodory theorem there exist a function continuous at satisfying

as is differentiable at a function continuous at satisfying

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**** is true for every taking we get

take then is continuous at as
is continuous at ,
is continues at and
is continuous at
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Theorem (Inverse of function).

Suppose are non-trivial intervels of and is a bijection, Take suppose is diffrentiatable at and is differentiable at
then
(in perticular )

Proof.
Take so
By Chain Rule Chain Rule

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Theorem.

Suppose I , are non-trivial intervals in and is abijection take suppose is diffrentiable at and suppose is Continuous at . then is differentiable at and

Proof.
By Carathéodory theorem Carathéodory theorem there exist a function continuous at satisfying

Claim.

for as is injective so
for ,

take then

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It is assumed that is continuous at and is continuous at so byCarathéodory theorem Carathéodory theorem is differentiable

Local Maxima Minima Local Maxima Minima