Differential calculus studies rates; integrals study totals
Differential calculus focuses on derivatives, differentials, and how fast quantities change. Paired with integral calculus through the fundamental theorem, it turns secant slopes into tangent slopes and simplifies powers with the familiar nx-to-the-n-minus-one rule.
The derivative at a point equals instantaneous rate of change and the slope of the tangent line. Physics reads position's derivative as velocity and velocity's as acceleration. Elsewhere derivatives locate maxima and minima and seed differential equations that model nature.
For y equals f of x, the derivative is the limit of a difference quotient as the step goes to zero when the limit exists; that quotient is a secant slope approaching the tangent. Example: if f of x is x squared, the derivative is 2x, so at x equals 2 the parabola's slope is 4.
Rather than grinding through limits every time, students lean on the sum, product, quotient and chain rules. The derivative also serves as the coefficient of the best linear approximation near a point: close to a, f of x is roughly f of a plus the slope times the distance from a, with an error that shrinks faster than that distance. That linear slice of the change is called the differential, written with dx for a small nudge to the input and dy for the matching estimated shift in output. Force in Newton's second law is the time derivative of momentum, and chemists use derivatives to express reaction rates. For functions of several variables, the same thinking yields partial, directional and total derivatives, while generalizations of the derivative turn up in abstract algebra, measure theory, differential geometry, and both complex and functional analysis.
Source: Differential calculus
