Derivative is a quantity that describes the local change speed of a function in mathematics. The derivative of a function can be defined by the rate of change of the function. For the function y=f(x), if the value of the function y changes with the change of the independent variable x, then when x becomes x+Δ x, the value of the function y will also change Δ y = f (x+Δ x)-f (x). The core of derivative is the rate of change, which describes the rate of change of a function at a certain point.
Derivatives are widely used in mathematics and other scientific fields. For example, in geometry, derivatives can be used to describe the slope of curves; In physics, derivatives can describe speed and acceleration; In economics, derivatives can be used to analyze costs, benefits and profits.
There are limits, definitions and formulas for calculating derivatives. Finding the limit is the most basic derivative calculation method, which is based on the definition of function change rate. The definition method is to calculate the derivative through the definition of derivative. Formula rule is to use some common derivative formulas to calculate derivatives.
The importance of learning derivatives;
1, Understanding Function Properties: Derivative is a key tool to study function properties. By analyzing the sign and magnitude of the derivative, we can understand the important properties of the function, such as monotonicity, extreme point, inflection point and maximum value. This helps us to understand and solve various practical problems.
2. Optimization: In real life, we often encounter problems that need optimization, such as finding the shortest route of the trip and determining the optimal distribution mode of materials. Derivative can help us understand and solve these problems, and provide the necessary conditions for the optimal solution.
3. Natural science research: In natural science research, derivatives are also widely used. For example, in physics, derivatives can describe the motion state of an object, the change of energy and the diffusion of matter. In chemistry, derivatives are also used to describe the dynamic process of chemical reactions.
4. Economic analysis: in economic analysis, derivatives are used to study and analyze the dynamic changes of the economy, such as price changes, demand fluctuations, investment returns, etc.
5. Computer science: In computer science, derivatives are also used in various optimization and decision-making, such as gradient descent algorithm of machine learning and edge detection of image processing.