Understanding Derivatives: From Rate of Change to Function Slopes

This article introduces the concept of derivatives and the derivatives of common functions.

1. Rate of Change

We first look at the rate of change. It can be calculated by dividing the difference in function values by the difference in the independent variable.

Here Δy represents the difference in function values and Δx represents the difference in the independent variable. A positive rate of change indicates the function value increases, while a negative rate indicates it decreases.

Rate of change is commonly used in physics. Imagine a car traveling on a highway; its position at time t₁ is x₁ and at time t₂ is x₂. The car's speed is defined as the ratio of the spatial difference to the time difference.

From the above expression we see this is the difference of the position function with respect to the time step Δt. This is only an approximation of the car's speed change. However, as the time step becomes very small, the approximation approaches the true value (since speed cannot change drastically over an infinitesimally small interval).

2. Definition of Derivative: Limit of Rate of Change

Physics also has the concept of instantaneous rate of change. We define the instantaneous rate of change of a function as the limit of the rate of change.

This limit involves an indeterminate form because the numerator and denominator both approach zero. However, this limit often exists, and we call the resulting function the derivative of f. The derivative gives the instantaneous rate of change at each point.

3. Geometric Interpretation of Derivatives: Slope of the Tangent

The geometric interpretation of the derivative is the slope of the tangent line. If we have a function f and two points (x₁, f(x₁)) and (x₂, f(x₂)), there is a line intersecting the function at those Cartesian coordinates.

The slope of this line is the average rate of change of the function. As the interval becomes smaller, the two points get closer. In the limit as the interval approaches zero, the graph of the function and this line locally touch at a single point. This limiting line is the tangent to the curve near that point, and its slope is the derivative.

We consider how to find the best linear approximation near a point. This linear function should have the form:

In fact, the slope of the line is the derivative at the point of tangency.

4. Derivatives of Important Functions

4.1 Constant Function

Proof omitted.

4.2 Linear Function

Proof omitted.

4.3 Quadratic Function

Proof omitted.

4.4 Power Function

Proof uses the binomial theorem. Detailed proof omitted.

4.5 Exponential Function

Proof based on properties of the exponential function. Detailed proof omitted.

4.6 Sine Function

Proof omitted.

4.7 Cosine Function

Proof omitted.

5. Summary

This article introduced the definition of the derivative, its physical meaning (instantaneous rate of change), geometric meaning (slope), and briefly presented and proved the derivatives of common functions.