Differentiability

• Differentiability = if there exists a best possible linear approximation
• Because partial derivatives do not need a function to be continuous, partial derivatives existing does not mean a function is differentiable

Hard Version

$f$ is differentiable if the following limit is true:

$$\underset{P\to P_0}{\lim }\frac{|f(P)-L(P)|}{|P{P}_0|}=0$$

While this is hard to use, we can use it to rule out functions that are

• Not continuous at $P_0$
• A partial derivative DNE at $P_0$

Not Pain Version

• All partial derivatives are continuous $⟹f$ is differentiable
• The converse is not true (sufficient but not necessary)