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Limit Definition of Derivative

The limit definition of derivative is a formula that gives the exact slope of a curve at a single point by taking the limit of a difference quotient as the interval shrinks to zero. It is the foundational definition from which all derivative rules are derived.

The derivative of a function ff at a point xx is defined as the limit of the difference quotient f(x+h)−f(x)h\frac{f(x+h) - f(x)}{h} as hh approaches zero, provided this limit exists. Geometrically, this represents the slope of the tangent line to the graph of ff at the point (x,f(x))(x, f(x)). When the limit does not exist, the function is said to be non-differentiable at that point. An equivalent form replaces hh with (a−x)(a - x) and takes the limit as a→xa \to x.

Key Formula

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
Where:
  • f′(x)f'(x) = the derivative of f at x
  • f(x)f(x) = the original function evaluated at x
  • hh = a small increment that approaches zero

Worked Example

Problem: Use the limit definition to find the derivative of f(x)=x2+3xf(x) = x^2 + 3x.
Step 1: Write out the difference quotient by computing f(x+h)f(x+h).
f(x+h)=(x+h)2+3(x+h)=x2+2xh+h2+3x+3hf(x+h) = (x+h)^2 + 3(x+h) = x^2 + 2xh + h^2 + 3x + 3h
Step 2: Subtract f(x)f(x) from f(x+h)f(x+h).
f(x+h)−f(x)=(x2+2xh+h2+3x+3h)−(x2+3x)=2xh+h2+3hf(x+h) - f(x) = (x^2 + 2xh + h^2 + 3x + 3h) - (x^2 + 3x) = 2xh + h^2 + 3h
Step 3: Divide by hh to form the difference quotient.
f(x+h)−f(x)h=2xh+h2+3hh=2x+h+3\frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2 + 3h}{h} = 2x + h + 3
Step 4: Take the limit as h→0h \to 0.
f′(x)=lim⁡h→0(2x+h+3)=2x+3f'(x) = \lim_{h \to 0}(2x + h + 3) = 2x + 3
Answer: The derivative is f′(x)=2x+3f'(x) = 2x + 3.

Why It Matters

The limit definition is the theoretical foundation of all of calculus. Every shortcut rule you learn — the power rule, the product rule, the chain rule — can be proven starting from this definition. In physics and engineering, understanding this definition clarifies what an instantaneous rate of change actually means, whether you're measuring velocity, current, or growth rate at a precise moment in time.

Common Mistakes

Mistake: Forgetting to expand f(x+h)f(x+h) fully before subtracting f(x)f(x).
Correction: You must substitute (x+h)(x+h) everywhere xx appears in the function and expand completely. Missing a term — especially from squaring a binomial — leads to an incorrect derivative.
Mistake: Plugging in h=0h = 0 before simplifying, which creates 00\frac{0}{0}.
Correction: The whole point is to simplify the fraction algebraically so that hh cancels from the denominator first. Only then do you let h→0h \to 0.

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