Limit Definition of the Derivative

Core Idea

The derivative of f at x = a is defined as f'(a) = lim(h->0) (f(a + h) - f(a))/h, the limit of the difference quotient. This single formula captures the instantaneous rate of change by taking the average rate of change over a shrinking interval. When this limit exists, the function is said to be differentiable at a. Every derivative rule you will learn is a shortcut derived from this definition.

How It's Best Learned

Compute derivatives from the definition for simple functions: f(x) = x², f(x) = 1/x, f(x) = sqrt(x). Show the algebra step by step, emphasizing how the h in the denominator cancels. Connect each computation to the slope of the tangent line. Then motivate the need for shortcut rules (the definition is correct but slow).

Common Misconceptions

• Setting h = 0 instead of taking the limit as h approaches 0 (division by zero).
• Forgetting to expand (a + h)ⁿ correctly.
• Believing the derivative exists everywhere just because the function is defined everywhere (absolute value at 0 is a counterexample).

Explainer

You already understand what a limit is: a number that an expression approaches as the input approaches some target value. And from your study of rates of change, you know that the average rate of change of f over an interval [a, a+h] is the slope of the secant line through the two points — computed as [f(a+h) − f(a)] / h. The derivative makes this exact by asking: what does this slope approach as the interval shrinks to nothing?

This expression [f(a+h) − f(a)] / h is called the difference quotient. For any fixed nonzero h, it gives the slope of the secant line through (a, f(a)) and (a+h, f(a+h)). As h → 0, the second point slides toward the first along the curve, the secant rotates, and in the limit it becomes the tangent line at (a, f(a)). The derivative f'(a) is exactly this limiting slope: f'(a) = lim(h→0) [f(a+h) − f(a)] / h. This is not a shortcut or an approximation — it is the definition. Every derivative rule you will learn (power rule, product rule, chain rule) is derived from this single formula.

Computing from the definition requires algebra. For f(x) = x², start with the difference quotient: [(a+h)² − a²] / h. Expand the numerator: [a² + 2ah + h² − a²] / h = [2ah + h²] / h. Factor out h: h(2a + h) / h = 2a + h. Now take the limit as h → 0: the result is 2a. The critical step is the cancellation of h from numerator and denominator — which is *why* you cannot substitute h = 0 at the start. Substituting h = 0 initially gives 0/0, an indeterminate form that yields no information. The limit process first simplifies the algebra until h cancels, then evaluates at h = 0.

Not every function has a derivative everywhere. Differentiability requires the limit to exist, which demands the function behave smoothly at that point — no sharp corners, no vertical tangents, no breaks. The function |x| is continuous everywhere but fails to be differentiable at x = 0: approaching from the left, the secant slope approaches −1; from the right, it approaches +1. The one-sided limits disagree, so the limit does not exist. Differentiability is a strictly stronger condition than continuity: every differentiable function is continuous, but not every continuous function is differentiable. Knowing this distinction will prevent errors when you apply derivative rules without checking whether they apply.