What Is a Derivative? Master Rate of Change, Tangent Slopes, and Core Functions

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

1 Rate of Change

The rate of change of a function is computed by dividing the difference in function values by the difference in the independent variable. A positive rate indicates an increase, while a negative rate indicates a decrease. In physics, this concept describes instantaneous speed, such as a car’s position change over time.

2 Derivative Definition: Limit of the Rate of Change

The instantaneous rate of change is defined as the limit of the average rate as the interval approaches zero. This limit, when it exists, is called the derivative of the function and represents the function’s instantaneous change at each point.

3 Geometric Interpretation: Slope of the Tangent

The derivative can be viewed geometrically as the slope of the tangent line to the graph of the function at a point. As the two points used in the difference quotient become arbitrarily close, the secant line approaches the tangent line, whose slope equals the derivative.

4 Derivatives of Important Functions

4.1 Constant Function

The derivative of a constant function is zero.

4.2 Linear Function

The derivative of a linear function f(x)=mx+b is the constant m.

4.3 Quadratic Function

The derivative of f(x)=ax^2+bx+c is f'(x)=2ax+b.

4.4 Power Function

Using the binomial theorem, the derivative of f(x)=x^n is f'(x)=n·x^{n‑1}.

4.5 Exponential Function

By the properties of exponential functions, the derivative of f(x)=a^x is f'(x)=a^x·ln(a).

4.6 Sine Function

The derivative of f(x)=sin(x) is cos(x).

4.7 Cosine Function

The derivative of f(x)=cos(x) is –sin(x).

5 Summary

The article presented the definition of a derivative, its physical meaning as an instantaneous rate of change, its geometric meaning as a tangent slope, and gave brief proofs for the derivatives of several common functions.