The Derivative as a Function

Definition

The derivative function $f'$ is a new function whose output at any $x$ is the slope of the original function $f$ at that same $x$.

Why It Matters

Treating the derivative as a function allows us to analyze the continuous evolution of change. It provides a complete map of a system’s dynamics, enabling us to identify every moment of peak performance, decline, or stability across its entire range.

Core Concepts

• Functional Definition: $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$
  • How to read: “The derivative f prime of x equals the limit as h approaches zero of the ratio of the quantity f of x plus h minus f of x to h.”
  • Meaning / when to use: Defines the derivative at every point $x$ where the limit exists—turns a pointwise slope into a function $f'$.
• Domain: The domain of $f'$ consists of all points in the domain of $f$ where the function is differentiable.
• Differentiation: The operation of transforming $f$ into $f'$.

Connected Concepts

• Derivative as a Rate of Change
• Derivative at a Point
• Continuity at a Point