Difference quotient

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In mathematics, for a real valued function of a single variable, the difference quotient measures the average rate of change of that function over some interval. The difference quotient is used in defining derivatives, by taking the limit of the difference quotient as the size of the interval approaches zero. In differential equations, the difference quotient is used as an approximation of the derivative of a function, and is used to provide numerical approximations of solutions of differential equations. There are many variations of the difference quotient throughout advanced calculus and analysis, including versions for vector valued functions, Fréchet Derivatives (limits of quotients of norms of vector valued differences), a complex difference quotient, and so on. [math]f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}[/math]