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Derivative vs. Integral — Definition, Formula & Examples

A derivative measures the instantaneous rate of change of a function, while an integral measures the accumulated total (area) under a function. They are inverse operations: differentiation undoes integration, and integration undoes differentiation.

Given a continuous function ff on [a,b][a, b], the derivative f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} yields the slope of ff at each point, whereas the definite integral ∫abf(x) dx=lim⁡n→∞∑i=1nf(xi∗) Δx\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*)\,\Delta x yields the signed area between ff and the xx-axis. The Fundamental Theorem of Calculus establishes that if F′(x)=f(x)F'(x) = f(x), then ∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b) - F(a).

Key Formula

ddx ⁣∫axf(t) dt=f(x)\frac{d}{dx}\!\int_a^x f(t)\,dt = f(x)
Where:
  • f(t)f(t) = A continuous function being integrated
  • aa = Lower bound of integration (a constant)
  • xx = Upper bound of integration, treated as the variable

How It Works

Think of a derivative as zooming in on one instant — it tells you how fast something is changing right now. An integral zooms out and adds up all the tiny changes over an interval. If you differentiate a position function s(t)s(t), you get velocity v(t)v(t). If you integrate that velocity back, you recover the change in position. This inverse relationship is precisely what the Fundamental Theorem of Calculus states: differentiation and integration undo each other, up to a constant.

Worked Example

Problem: Let f(x)=3x2f(x) = 3x^2. Find its derivative and then integrate the result from 0 to 2 to verify the inverse relationship.
Differentiate: Apply the power rule to find the derivative of f(x)=3x2f(x) = 3x^2.
f′(x)=6xf'(x) = 6x
Integrate the derivative: Integrate f′(x)=6xf'(x) = 6x from 0 to 2 using the power rule for integrals.
∫026x dx=3x2∣02=3(4)−3(0)=12\int_0^2 6x\,dx = 3x^2\Big|_0^2 = 3(4) - 3(0) = 12
Verify with the original function: Compute f(2)−f(0)f(2) - f(0) directly to confirm the Fundamental Theorem.
f(2)−f(0)=3(4)−3(0)=12f(2) - f(0) = 3(4) - 3(0) = 12
Answer: Both methods give 12, confirming that integrating the derivative recovers the net change in the original function.

Why It Matters

Understanding how derivatives and integrals relate is essential for solving differential equations in physics and engineering. In any applied calculus or dynamics course, switching between rate-of-change problems and accumulation problems is a daily task.

Common Mistakes

Mistake: Forgetting the constant of integration when finding an indefinite integral, then claiming the integral perfectly "reverses" the derivative.
Correction: An indefinite integral ∫f′(x) dx=f(x)+C\int f'(x)\,dx = f(x) + C. The constant CC is lost during differentiation, so you must include it when integrating. Definite integrals avoid this issue because the constant cancels in F(b)−F(a)F(b) - F(a).