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URL: https://nakafa.com/en/subjects/mathematics/derivative-function/concept-of-derivative-function
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/derivative-function/concept-of-derivative-function/en.mdx
Understand derivative functions as rates of change and curve slopes through limits, tangent lines, and visual examples.
---
## The Idea Behind Derivatives
Imagine you're riding a bicycle on a hilly road. Sometimes the road is steep, and other times it's flat. The **slope** of the road changes at every point you pass. In mathematics, the graph of a function can be thought of as this hilly road.
For a straight line, the slope is always the same at every point. However, for a curved line, the slope is constantly changing. A **derivative** lets us find the precise slope or rate of change at **one specific point** on a curve.
## Gradient of a Secant Line
To understand the concept of a derivative, let's start with something simpler: a **secant line** (or a cutting line). A secant line is a straight line that intersects a curve at two different points.
Suppose we have a curve from the function $$y = f(x)$$. We pick two points on that curve, let's call them point $$P(x, f(x))$$ and point $$Q(x+\Delta x, f(x+\Delta x))$$. Here, $$\Delta x$$ (read "delta x") represents a small change in the value of $$x$$.
Visible text: Suppose we have a curve from the function . We pick two points on that curve, let's call them point and point . Here, (read "delta x") represents a small change in the value of .
The slope (gradient) of the secant line passing through points $$P$$ and $$Q$$ can be calculated with a formula we already know:
Visible text: The slope (gradient) of the secant line passing through points and can be calculated with a formula we already know:
```math
m_{\text{secant}} = \frac{\text{change in } y}{\text{change in } x} = \frac{f(x + \Delta x) - f(x)}{\Delta x}
```
The gradient of this secant line gives us an idea of the **average rate of change** of the function $$f(x)$$ between points $$P$$ and $$Q$$.
Visible text: The gradient of this secant line gives us an idea of the **average rate of change** of the function between points and .
Component: LineEquation
Props:
- title: Secant and Tangent Line Visualization
- description: Notice how the secant line connects two points on the curve{" "}
$$y=x^2$$, while the tangent line just
touches the curve at a single point. The tangent line shows the slope of the curve
at that point.
Visible text: Notice how the secant line connects two points on the curve{" "}
, while the tangent line just
touches the curve at a single point. The tangent line shows the slope of the curve
at that point.
- showZAxis: false
- cameraPosition: [0, 0, 15]
- data: (() => {
// Define the curve function
const f = (x) => x * x;
// 1. Define the main curve (parabola y = x^2)
const curvePoints = Array.from({ length: 101 }, (_, i) => {
const x = (i - 50) / 10; // x from -5 to 5
return { x, y: f(x), z: 0 };
});
// 2. Define the secant line
const p1_secant = { x: 1, y: f(1), z: 0 };
const p2_secant = { x: 3, y: f(3), z: 0 };
// 3. Define the tangent line at point P
const tangentPointX = 1;
const tangentPoint = { x: tangentPointX, y: f(tangentPointX), z: 0 };
const slope = 2 * tangentPointX; // Derivative of x^2 is 2x
// Line equation: y - y1 = m(x - x1) => y = m(x - x1) + y1
const tangentLineFunc = (x) => slope * (x - tangentPointX) + tangentPoint.y;
const tangentLinePoints = [
{ x: -1, y: tangentLineFunc(-1), z: 0 },
{ x: 3, y: tangentLineFunc(3), z: 0 },
];
return [
{
points: curvePoints,
color: getColor("PURPLE"),
showPoints: false,
},
{
points: [p1_secant, p2_secant],
color: getColor("CYAN"),
labels: [
{ text: "P", at: 0, offset: [-0.5, -0.5, 0] },
{ text: "Q", at: 1, offset: [0.5, 0.5, 0] },
{ text: "Secant Line", at: 0, offset: [-1, 2.5, 0] },
],
},
{
points: tangentLinePoints,
color: getColor("AMBER"),
showPoints: false,
labels: [{ text: ... [truncated; 1258 chars]
## From Secant Line to Tangent Line
Now, what happens if we move point $$Q$$ closer and closer to point $$P$$? The distance between them, which is $$\Delta x$$, will become very small, approaching zero.
Visible text: Now, what happens if we move point closer and closer to point ? The distance between them, which is , will become very small, approaching zero.
When $$\Delta x \to 0$$ (read "delta x approaches zero"), the secant line we have will gradually transform into a **tangent line**. A tangent line is a line that touches the curve at exactly one point (in this case, point $$P$$).
Visible text: When (read "delta x approaches zero"), the secant line we have will gradually transform into a **tangent line**. A tangent line is a line that touches the curve at exactly one point (in this case, point ).
The slope of this tangent line is what truly represents the **slope of the curve** at point $$P$$. To find it, we use the concept of a **limit**.
Visible text: The slope of this tangent line is what truly represents the **slope of the curve** at point . To find it, we use the concept of a **limit**.
```math
m_{\text{tangent}} = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}
```
## Definition of the Derivative
The limit of the gradient of the secant line as $$\Delta x$$ approaches zero is so important that it is given a special name: the **derivative**.
Visible text: The limit of the gradient of the secant line as approaches zero is so important that it is given a special name: the **derivative**.
The derivative of a function $$f(x)$$, denoted as (read "f prime x"), is defined as:
Visible text: The derivative of a function , denoted as (read "f prime x"), is defined as:
```math
f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}
```
The process of finding this derivative is called **differentiation**.
> The derivative is essentially a new function that tells us the **instantaneous rate of change** (or the slope of the tangent line) of the original function $$f(x)$$ at every point $$x$$ where the limit exists. This is the foundation of differential calculus.
Visible text: > The derivative is essentially a new function that tells us the **instantaneous rate of change** (or the slope of the tangent line) of the original function at every point where the limit exists. This is the foundation of differential calculus.