Differentiability

A function is differentiable at iff:

When it is differentiable, is called the derivative of at .

Critical point

is called a critical point iff:

One-side differentiable

is differentiable at iff is left differentiable at and is right differentiable at .

Left differentiable

A function is left-differentiable at iff:

Right differentiable

A function is right-differentiable at iff:

On intervals

Open interval

A function is differentiable in iff is differentiable on every .

Closed interval

A function is differentiable in iff is:

• differentiable on every
• right-differentiable at
• left-differentiable at

Continuously differentiable functions

A function is said to be continously differentiable at iff :

• is differentiable at and
• is continous at

Differentiability implies continuity

Likewise, one-sided differentiability implies corresponding one-sided continuity.

Proof Hint

Use .

Note

Suppose is differentiable at . Define :

is continuous at .

Properties of differentiation

Addition

Multiplication

Division

Composition

Power