Differentiation rules

Differentiation rules are formulas for finding derivatives of sums, products, quotients, compositions, and common functions. They turn derivative calculations into reusable steps.

Connect

Derivative: A derivative measures a function’s instantaneous rate of change and, geometrically, the slope of its tangent. Every differentiation rule computes this quantity from a function’s structure.
Sum rule: The sum rule states that the derivative of a sum is the sum of the derivatives. It handles addition and subtraction term by term.
Exponential function: An exponential function has a constant base raised to a variable exponent; the natural exponential is its principal calculus example. Its derivative is especially simple for the natural base and follows by composition for other bases.
Higher-order derivative: A higher-order derivative is obtained by differentiating a function’s derivative one or more times. Repeated application of differentiation rules produces curvature and other rate-of-change information.
Limit of a function: A limit describes the value a function approaches as its input approaches a specified point. The derivative’s definition uses a limit of average rates of change.
Product rule: The product rule gives the derivative of a product as the first factor times the second derivative plus the second factor times the first derivative. It accounts for both factors changing at once.
Logarithmic function: A logarithmic function gives the exponent to which a base must be raised to produce its input. Its derivative formula also enables logarithmic differentiation of complicated products and powers.
Implicit function theorem: The implicit function theorem gives conditions under which an equation locally defines one variable as a differentiable function of others. Its derivative formula relies on differentiating the defining equation.
Differentiability: Differentiability is the property of having a derivative at a point or throughout a domain. Rules require differentiable inputs wherever their formulas are applied.
Quotient rule: The quotient rule gives the derivative of a ratio from the derivatives and values of its numerator and denominator. It handles division when the denominator is nonzero.