Miscellaneous Proofs
where
This creates the following inequality:
The base,
For the height,
Then,
Then,
Therefore, we have
Since
Multiplying through by
Taking reciprocals gives
For
It follows that
Hence,
Derivative of a constant
[!abstract]
for
Let
Derivative of function times a constant
[!abstract]+
Letand be a differentiable function, then
Let
Derivative of
Derivative of
Derivative of
Derivative of
[!abstract]
Note that
Let
Derivative of
Derivative of
Differentiability implies continuity
We have to show that
is certain to exist.
Then,
The contrapositive of the theorem says non-continuity $\implies$ non-differentiability.
The converse is not necessarily true, i.e. continuity doesn't imply differentiability.