1, (u/v)' = (u' v-uv')/(v 2): This formula is the derivative formula of quotient, where u and v are two functions. This formula shows that when the quotient of two functions u and v changes, the change rate is equal to the derivative of u times v minus the derivative of v times u and then divided by the square of v. This formula is very useful in calculating the derivative of composite function.
2.(uv)'=u'v+uv': This formula is the derivative formula of product, where u and v are two functions. This formula shows that when two functions u and v are multiplied, their derivatives are equal to the derivatives of u times v plus the derivatives of v times u. This formula is very useful in calculating the products of multiple functions.
3.(f(g(x))' = f' (g (x)) g' (x): This formula is the derivative formula of composite function, where f and g are two functions, and f(g(x)) is a new function composed of f and g. This formula shows that when one function F is compounded with another function G, its derivative is equal to that of F.
Application scenarios of mathematical derivatives;
1. Optimization problem: Derivative plays a key role in optimization problem. For example, when finding the minimum or maximum value of a function, we can find the point where the derivative is zero to find the extreme point by finding the derivative. In engineering, economy, finance and other fields, optimization problems are very common, and this application scenario of derivative is very important.
2. Curve fitting: In science and engineering, curves are often needed to fit data. Derivative can be used to determine the shape and properties of curves, such as linear regression and quadratic regression. In signal processing and other fields, derivatives are also used to analyze and process function curves.
3. Control system: In the control system, the derivative is used to describe the response and stability of the system. For example, in mechanical engineering, by using derivatives to analyze the dynamic characteristics of mechanical systems, the performance of the system can be optimized and unstable situations can be avoided. In addition, derivatives are also used in the design and analysis of control systems in aerospace and automobile fields.