Derivative

analysis

Definition

Derivative

The limit of a differentiable function at is called the derivative of at point and is commonly denoted as .

Notations:

• Leibniz Notation:
• Limit Notation:
• Newton Notation:

Rules

Constant Factor Rule

Sum Rule

Together with the constant factor rule, differentiation is a linear mapping.

Product Rule

Quotient Rule

If , then:

Chain Rule

Let a composed function. Then:

The function is called the outer function and is called the inner function.

In Leibniz’s notation:

Inverse Rule

If is invertible and does not have any root points, then for all it holds that:

In Leibniz’s notation:

Common Derivative Functions

Constant Function

Let be a constant function. It holds that for all . Thus, the derivative of is .

Linear Function

Let be a linear function. The derivative is given by:

Let . The derivative is given by:

Power Function

Let , with , be a power function. The derivative is given by:

Absolute Function

Let be the absolute function

Therefore, we initially have:

However, the interesting case is . In this case:

Since takes on both the values and in every neighbourhood around , the limit does not exist. hence, although the function is continuous at , it is not differentiable here.

Sine Function

Let be the sine function. The derivative is given by:

Analogously, .

Natural Exponential Function

Let be the natural exponential function. The derivative is given by:

By substituting the exponential series, one can further rewrite it as:

This series has a radius of convergence . Hence, by the continuity of power series, it follows that

and therefore

Examples

Example:

1. :
2. :
3. :

The limit for is not defined, as there in every epsilon neighbourhood around , the function either becomes or . Therefore, only the left-sided limit and right-sided limit exist.