Derivatives: the basics of differential calculus

In short: The derivative of a function describes how strongly its value changes at a particular point, in other words its slope. It’s the central tool for mathematically capturing rates of change, like speed or growth.

In more detail: Geometrically, the derivative at a point corresponds to the slope of the tangent line to the function’s graph at that point.

In detail

• Difference quotient: The derivative arises as the limit of the difference quotient as the distance between two points approaches zero.
• Differentiation rules: The power rule, product rule, chain rule, and quotient rule allow derivatives to be calculated systematically instead of forming the limit anew each time.
• Slope and tangent: A positive derivative means an increasing function, a negative one a decreasing function, and a derivative of zero a possible maximum or minimum point.
• Speed as an example: The derivative of the distance-time function gives speed, and its derivative in turn gives acceleration.
• Finding extrema: Zeros of the first derivative indicate possible maxima or minima of a function.
• Second derivative: It describes the curvature of the function’s graph and helps distinguish between a maximum and a minimum.
• Practical application: From optimization problems in economics to physics, differential calculus is used to analyze rates of change (gleichungen-loesen-grundlagen).