Taylor formula differential equation: f(x)=f(x0)+f'(x0)(x0). Taylor formula, which is applied in mathematics, physics and other fields, is a formula that uses the information of a function at a certain point to describe the value near it. If the function is smooth enough, Taylor's formula can use these derivative values as coefficients to construct a polynomial to approximate the function value in the neighborhood of the point when the derivative values of the function are known. Taylor formula also gives the deviation of this polynomial from the actual function value.

Differential equations are developed by calculus. Newton and Leibniz, the founders of calculus, both dealt with problems related to differential equations in their works. Differential equations are widely used and can solve many problems related to derivatives. Many kinematics and dynamics problems involving variable forces in physics, such as falling bodies with air resistance as speed function, can be solved by differential equations. In addition, differential equations have applications in chemistry, engineering, economics and demography.