👉 To understand the math behind a simple yet powerful concept like the derivative, consider a function \( f(x) \) that describes a curve on a coordinate plane. The derivative of \( f(x) \), denoted \( f'(x) \), measures how the function changes as \( x \) changes. Mathematically, this is defined as the limit of the average rate of change of the function over an interval \([a, b]\) divided by the width of that interval: \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \). This limit essentially captures the instantaneous rate of change at any point \( x \) on the curve. Geometrically, it gives the slope of the tangent line to the curve at that point, providing critical insights into the function's behavior, such as identifying maxima, minima, and inflection points. This derivative is foundational in calculus, enabling us to analyze functions' rates of change and optimize various physical and economic phenomena.
