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[5] [6] The difference quotient is a measure of the average rate of change of the function over an interval (in this case, an interval of length h). [7] [8]: 237 [9] The limit of the difference quotient (i.e., the derivative) is thus the instantaneous rate of change. [9]
The process of finding the difference quotient is called differentiation. Given a function defined at several points of the real line, the difference quotient at that point is a way of encoding the small-scale (i.e., from the point to the next) behavior of the function.
Therefore, the true derivative of f at x is the limit of the value of the difference quotient as the secant lines get closer and closer to being a tangent line: ′ = (+) (). Since immediately substituting 0 for h results in 0 0 {\displaystyle {\frac {0}{0}}} indeterminate form , calculating the derivative directly can be unintuitive.
The latter is the difference quotient for g at a, and because g is differentiable at a by assumption, its limit as x tends to a exists and equals g′(a). As for Q(g(x)), notice that Q is defined wherever f is. Furthermore, f is differentiable at g(a) by assumption, so Q is continuous at g(a), by definition of the derivative.
The finite difference of higher orders can be defined in recursive manner as Δ n h ≡ Δ h (Δ n − 1 h) . Another equivalent definition is Δ n h ≡ [T h − I ] n . The difference operator Δ h is a linear operator, as such it satisfies Δ h [ α f + β g ](x) = α Δ h [ f ](x) + β Δ h [g](x) . It also satisfies a special Leibniz rule:
This expression is called a difference quotient. A line through two points on a curve is called a secant line, so m is the slope of the secant line between (a, f(a)) and (a + h, f(a + h)). The second line is only an approximation to the behavior of the function at the point a because it does not account for what happens between a and a + h.
For differentiable functions, the symmetric difference quotient does provide a better numerical approximation of the derivative than the usual difference quotient. [3] The symmetric derivative at a given point equals the arithmetic mean of the left and right derivatives at that point, if the latter two both exist. [1] [2]: 6
The derivative of the function at a point is the slope of the line tangent to the curve at the point. Slope of the constant function is zero, because the tangent line to the constant function is horizontal and its angle is zero. In other words, the value of the constant function, y, will not change as the value of x increases or decreases.