Derivatives

A derivative measures the rate at which a function changes with respect to its variable. In geometric terms, the derivative represents the slope of the tangent line to a curve at a given point. In physical terms, it can represent quantities such as velocity (rate of change of position) or acceleration (rate of change of velocity). The derivative is defined using a limit of a difference quotient. If f(x) is a function, its derivative is written as f'(x) or dy/dx. Derivatives help us determine where a function increases or decreases. They are also used to find maximum and minimum values, which is essential in optimization problems. In economics, derivatives are used to calculate marginal cost and marginal revenue. In science and engineering, they describe how systems evolve over time. Higher-order derivatives (like the second derivative) provide information about concavity and curvature of graphs.

(xⁿ)' = n·xⁿ⁻¹

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