Differentiation Theorems

Differentiability Implies Continuity

Theorem

If if differentiable at , then is continuous at

Proof

Observe that
Take the limit of both sides:

Thereby fulfilling the continuity criterion

Remark

The converse of this theorem is not necessarily true

Mean Value Theorem (MVT)

Theorem

If is continuous on and differentiable on , then there exists a number such that

Corollary

If for all , then is constant on

That is, if the derivative is 0, for a range, then the function has a constant rate of change.

Proof

Let and be two distinct points on . By MVT, there exists a number such that
But we've assumed that , so . Since and are arbitrary, has the same value everywhere in

L'Hôpital's Rule

Theorem

Consider a limit in the form , where and are differentiable on some open interval containing .

If and (so that the limit is of indeterminate form ), then , if this limit exists

The same result holds for the indeterminate form , and also holds if we use any other limit (i.e. for one-sided limits, as well as )

Note

This comes from the linear approximation!

Approximating