What derivatives and integrals are
A car that covers 120 km in two hours averages 60 km/h, but the speedometer shows the speed at this very instant. Differentiation finds that instantaneous rate of change by shrinking the time interval towards zero, using the idea of a limit; on a graph it is the slope of the tangent line. Differentiating a function f(x) gives its derivative f′(x); the derivative of x², for example, is 2x. Integration works the other way, stacking ever-thinner rectangles to find the area under a curve or an accumulated total; integrating a speed graph gives distance travelled. The idea goes back to Archimedes, who used the method of exhaustion to find areas of circles and parabolic segments.
The fundamental theorem, Newton and Leibniz
The fundamental theorem of calculus says differentiation and integration undo each other, so instead of summing slices you can evaluate a definite integral by finding an antiderivative. Isaac Newton developed his "method of fluxions" in 1665–66, while Cambridge was closed by plague, and used it for his laws of motion and gravitation. Gottfried Leibniz developed calculus independently and published first, in 1684; the dx and ∫ notation we still use is his. A bitter priority dispute followed between British and continental mathematicians, but historians now agree the two discovered it independently. The rigorous epsilon-delta definition of limits came in the 19th century from Cauchy and Weierstrass.
Where it is used
Calculus is the language of almost every field that deals with change: equations of motion, electrical circuits and fluid flow in physics; marginal cost and marginal revenue in economics; how quickly a drug is cleared from the body; models of population growth. The gradient descent and backpropagation that train neural networks rely on derivatives and the chain rule. In many school systems students meet derivatives and integrals of polynomials first, then of exponential, logarithmic and trigonometric functions, and first-year calculus is required in most science and engineering degrees.
FAQ
- Which comes first, derivatives or integrals?
- Derivatives are usually taught first, because most integrals are calculated as the reverse of differentiation, which makes more sense once you understand derivatives.
- Is calculus really used in everyday life?
- You may rarely do it by hand, but navigation speed estimates, weather forecasts, financial pricing, medical imaging and AI training all run on calculus-based models.
Sources reviewed: 2026-10-26
References
- Boyer, C. B. (1959). The History of the Calculus and Its Conceptual Development. Dover.
- Hall, A. R. (1980). Philosophers at War: The Quarrel between Newton and Leibniz. Cambridge University Press.
- Stewart, J. (2015). Calculus: Early Transcendentals (8th ed.). Cengage Learning.
- 교육부. (2023). 2028 대학입시제도 개편 확정안.