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"For the next generation"
© 2025 Astra.si. All rights reserved.
"For the next generation"
The derivative of a function is a key tool in differential calculus that allows for determining the rate of change of a function with respect to its variable. This process reveals fundamental properties of functions, such as the slopes of tangents, extrema (maximums and minimums), and inflection points. The following describes how to calculate the derivatives of a function f(x) and what the key aspects are.
The derivative of a function f(x) at a point x is defined as the limit of the ratio of the change in the function at that point as the interval of change, h, approaches 0. This is mathematically expressed as:
f′(x) = lim (as h→0) [f(x+h) – f(x)] / h
For a practical example, if we want to calculate the derivative of the function f(x) = x², we use the power rule to get f'(x) = 2x.
Derivatives allow for the determination of the rate of change of a function, which is useful in finding maximums, minimums, and analyzing the tangent to a curve. Derivatives are also the foundation for further concepts in mathematics, such as integrals and differential equations.
Knowledge of how to calculate the derivatives of a function is fundamental for the study and understanding of natural phenomena and technical applications. Understanding and using the basic rules of differentiation allows for deeper insights into the properties of functions and is indispensable when solving mathematical problems. Understanding derivatives opens the door to comprehending more complex concepts in mathematics and their practical application.
