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Difference quotients may also find relevance in applications involving Time discretization, where the width of the time step is used for the value of h. The difference quotient is sometimes also called the Newton quotient [10] [12] [13] [14] (after Isaac Newton) or Fermat's difference quotient (after Pierre de Fermat). [15]
The second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.
This expression is Newton's difference quotient (also known as a first-order divided difference). The slope of this secant line differs from the slope of the tangent line by an amount that is approximately proportional to h. As h approaches zero, the slope of the secant line approaches the slope of the tangent line.
The difference operator of difference equations is another discrete analog of the standard derivative. Δ f ( x ) = f ( x + 1 ) − f ( x ) {\displaystyle \Delta f(x)=f(x+1)-f(x)} The q-derivative, the difference operator and the standard derivative can all be viewed as the same thing on different time scales .
A finite difference is a mathematical expression of the form f (x + b) − f (x + a).If a finite difference is divided by b − a, one gets a difference quotient.The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems.
In principle, the derivative of a function can be computed from the definition by considering the difference quotient and computing its limit. Once the derivatives of a few simple functions are known, the derivatives of other functions are more easily computed using rules for obtaining derivatives of more complicated functions from simpler ones.
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The quotient / is not infinitely small; rather it is a real number. The use of infinitesimals in this form was widely criticized, for instance by the famous pamphlet The Analyst by Bishop Berkeley. Augustin-Louis Cauchy defined the differential without appeal to the atomism of Leibniz's infinitesimals.
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