MAT 241 LECTURE NOTE –**Ordinary differential equation**, in mathematics, an equation relating a function *f* of one variable to its derivatives. (The adjective *ordinary* here refers to those differential equations involving one variable, as distinguished from such equations involving several variables, called partial differential equations.)

The derivative, written *f*′ or *df*/*dx*, of a function *f* expresses its rate of change at each point—that is, how fast the value of the function increases or decreases as the value of the variable increases or decreases. For the function *f* = *ax* + *b* (representing a straight line), the rate of change is simply its slope, expressed as *f*′ = *a*.

For other functions, the rate of change varies along the curve of the function, and the precise way of defining and calculating it is the subject of differential calculus. In general, the derivative of a function is again a function, and therefore the derivative of the derivative can also be calculated, (*f*′)′ or simply *f*″ or *d*^{2}*f*/*dx*^{2}, and is called the second-order derivative of the original function. Higher-order derivatives can be similarly defined.

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