Calculus - The Concept of the Derivative (Differentiation from First Principles)

The Average Rate of Change is defined as the gradient of a secant line that passes through two points on a curve, $(x, f(x))$ and $(x+h, f(x+h))$. Visually, this is the slope of a straight line connecting two distinct locations on a graph, representing how much the function changes over the interval $h$.

The Instantaneous Rate of Change is the gradient of the curve at one specific point. Visually, this is the slope of the tangent line—a line that just touches the curve at that point. Differentiation is the mathematical tool used to calculate this exact slope.

Differentiation from First Principles is the formal algebraic process of finding the derivative by taking the limit of the secant gradient. As the horizontal distance $h$ between two points on a graph shrinks toward zero, the secant line rotates and eventually coincides with the tangent line at the point $x$.

The concept of the Limit ($h \to 0$) is central to calculus. In a visual sense, it describes the behavior of the gradient as the 'run' of our slope calculation becomes infinitely small. We cannot simply divide by zero, so we use the limit to observe the value the gradient approaches as $h$ vanishes.

Derivative Notation: The derivative of a function $f(x)$ is commonly written as $f'(x)$ or $\frac{dy}{dx}$. Geometrically, if you were to plot $f'(x)$, the $y$-value of this new graph at any point $x$ would represent the steepness (slope) of the original function $f(x)$ at that same $x$ coordinate.

Tangents and Normals: The tangent is a line with the gradient $m = f'(x)$. The normal is a line perpendicular to the tangent at the point of contact. Visually, the normal line stands at a $90^{\circ}$ angle to the curve; its gradient is the negative reciprocal of the tangent's gradient, calculated as $-\frac{1}{f'(x)}$.