# Rates of change

The difference quotient is the average rate of change of a function between two points:

$\frac{\Delta&space;f(t)}{\Delta&space;t}=&space;\frac{f(t_{1})-f(t_{0})}{t_{1}-t_{0}}.$

The instantaneous rate of change is the limit of the difference quotient as t1 and t0 approach each other, which is the derivative of f(t) at that point, denoted by f′(t). Derivatives are added by arithmetic addition, i.e., if f(t) = g(t) + h(t), then f′(t) = g′(t) + h′(t).

The independent variable of the difference quotient is the variable in the denominator. However, there is another kind of difference quotient in which the variable in the numerator is the independent variable:

$\frac{\Delta&space;x}{\Delta&space;g(x)}=&space;\frac{x_{1}-x_{0}}{g(x_{1})-g(x_{0})}=&space;\left&space;(\frac{g(x_{1})-g(x_{0})}{x_{1}-x_{0}}&space;\right&space;)^{-1}=&space;\left&space;(\frac{\Delta&space;g(x)}{\Delta&space;x}&space;\right&space;)^{-1}.$

This is the reciprocal difference quotient. Since the limit of a quotient is the quotient of the limits (provided the latter is well-defined), the limit of the reciprocal difference quotient as x1 and x0 approach each other is the reciprocal derivative of g(x) at that point, denoted by ‵g(x). Reciprocal derivatives are added by reciprocal arithmetic, i.e., if f(t) = g(t) + h(t), then ‵f(t) = ‵g(t) ⊕ ‵h(t).

These difference quotients may be generalized to a vector function in the numerator. The vector difference quotient is the average rate of change of a vector function between two points:

$\frac{\Delta&space;\mathbf{f}(t)}{\Delta&space;t}=&space;\frac{\mathbf{f}(t_{1})-\mathbf{f}(t_{0})}{t_{1}-t_{0}}.$

The limit of this expression in which t1 and t0 approach each other is the vector derivative at that point, denoted by f′(t). Vector derivatives are added by vector arithmetic addition, i.e., if f(t) = g(t) + h(t), then f′(t) = g′(t) + h′(t).

The vector reciprocal difference quotient is the reciprocal expression:

$\frac{\Delta&space;{\mathbf{x}}}{\Delta&space;g(x)}=&space;\frac{\mathbf{x}_{1}-\mathbf{x}_{0}}{g(x_{1})-g(x_{0})}=&space;\left&space;(\frac{g(x_{1})-g(x_{0})}{\mathbf{x}_{1}-\mathbf{x}_{0}}&space;\right&space;)^{-1}=&space;\left&space;(\frac{\Delta&space;g(x)}{\Delta&space;\mathbf{x}}&space;\right&space;)^{-1}.$

Since the limit of a quotient is the quotient of the limits (provided the latter is well-defined), the limit of this expression in which g(x1) and g(x0) approach each other is the vector reciprocal derivative at that point, denoted by ‵g(x). Vector reciprocal derivatives are added by vector reciprocal addition, i.e., if f(t) = g(t) + h(t), then f(t) = g(t) ⊕ h(t).