Derivative of the function f(x)

• categories: Real Analysis, Definition

The derivative of a function $f(x)$ measures how the function’s output changes as its input changes. It represents the rate of change or slope of the function at any given point.

Definition

The derivative of $f(x)$ at a point $x$ is defined as the limit of the difference quotient:

$$f^{\prime }(x)=\underset{\Delta x\to 0}{\lim }\frac{f(x+\Delta x)-f(x)}{\Delta x}$$

This formula gives the instantaneous rate of change of the function at the point $x$.

Notation

The derivative of $f(x)$ can be written in several different notations:

• $f^{\prime }(x)$
• $\frac{df}{dx}$
• $\frac{d}{dx}f(x)$
• $D_{x}f(x)$

Geometric Interpretation

The derivative at a point $x$ is the slope of the tangent line to the graph of $f(x)$ at that point. If the derivative is positive, the function is increasing at that point; if it’s negative, the function is decreasing