Differentiability definitions

Differentiability Definition: A function $f(x)$ is differentiable at a point $c$ if its derivative exists at that point. The derivative is the limit of...

Differentiability Definition:

A function $f(x)$ is differentiable at a point $c$ if its derivative exists at that point. The derivative is the limit of the difference quotient as it approaches 0:

$f'(c) = \lim_{h \to 0} \frac{f(c + h) - f(c)}{h}$

The derivative exists if this limit exists and is finite. If the limit exists and is finite, we say that $f(x)$ is differentiable at $c$ .

Key Points:

A function is differentiable at a point $c$ if it has a defined derivative at that point.
The derivative is a function that gives the instantaneous rate of change of the original function at a given point.
The derivative can be calculated by taking the limit of the difference quotient.
The derivative exists at all points where the function is defined.
A function is differentiable if its derivative exists at all points in its domain.

Examples:

A function $f(x) = x^2$ is differentiable at all points, since the derivative of $x^2$ is $2x$ .
A function $f(x) = \sin(x)$ is differentiable at all points, since the derivative of $\sin(x)$ is $\cos(x)$ .
A function $f(x) = 0$ is not differentiable at 0, since the derivative of 0 is undefined