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The derivative of a function of a real variable measures the sensitivity to change of a quantity (a function value or dependent variable) which is determined by another quantity (the independent variable). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time is advanced. The derivative of a function of a single variable at a chosen input value is the slope of the tangent line to the graph of the function at that point. This means that it describes the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives may be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector. The process of finding a derivative is called differentiation. The reverse process is called ''antidifferentiation''. The fundamental theorem of calculus states that antidifferentiation is the same as integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.〔Differential calculus, as discussed in this article, is a very well established mathematical discipline for which there are many sources. Almost all of the material in this article can be found in Apostol 1967, Apostol 1969, and Spivak 1994.〕 ==Differentiation and derivative== ''Differentiation'' is the action of computing a derivative. The derivative of a function of a variable is a measure of the rate at which the value of the function changes with respect to the change of the variable. It is called the ''derivative'' of with respect to . If and are real numbers, and if the graph of is plotted against , the derivative is the slope of this graph at each point. The simplest case, apart from the trivial case of a constant function, is when is a linear function of , meaning that the graph of divided by is a line. In this case, , for real numbers and , and the slope is given by : where the symbol (Delta) is an abbreviation for "change in." This formula is true because : Thus, since : it follows that : This gives an exact value for the slope of a line. If the function is not linear (i.e. its graph is not a straight line), however, then the change in divided by the change in varies: differentiation is a method to find an exact value for this rate of change at any given value of . The idea, illustrated by Figures 1 to 3, is to compute the rate of change as the limit value of the ratio of the differences as becomes infinitely small. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「derivative」の詳細全文を読む スポンサード リンク
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