微分係数:Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point; for example, the derivative of the position (or distance) of a vehicle with respect to time is the instantaneous velocity (respectively, instantaneous speed) at which the vehicle is traveling. Conversely, the integral of the velocity over time is the vehicle's position. The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point.

微分と微分係数

Differentiation is a method to compute the rate at which a dependent output y changes with respect to the change in the independent input x. This rate of change is called the derivative of y with respect to x. In more precise language, the dependence of y upon x means that y is a function of x. If x and y are real numbers, and if the graph of y is plotted against x, the derivative measures the slope of this graph at each point. This functional relationship is often denoted y = ƒ(x), where ƒ denotes the function.

Differentiation is a method to find an exact value for this rate of change at any given value of x.

The idea is to compute the rate of change as the limiting value of the ratio of the differences Δy / Δx as Δx becomes infinitely small.

In Leibniz's notation, such an infinitesimal change in x is denoted by dx, and the derivative of y with respect to x is written

dy/dx

suggesting the ratio of two infinitesimal quantities.

The above expression is read as "the derivative of y with respect to x", "d y by d x", or "d y over d x". The oral form "d y d x" is often used conversationally, although it may lead to confusion.

導関数:Derivative of the function

The derivative is the value of the difference quotient as the secant lines approach the tangent line. Formally, the derivative of the function ƒ at a is the limit

derivative.png

of the difference quotient as h approaches zero, if this limit exists. If the limit exists, then ƒ is differentiable at a.

表記法:notation

  • Leibniz's notation The notation for derivatives introduced by Gottfried Leibniz is one of the earliest. It is still commonly used when the equation y = ƒ(x) is viewed as a functional relationship between dependent and independent variables. Then the first derivative is denoted by
    dy/dx,df/dx or d/dxf(x).
  • Lagrange's notation Sometimes referred to as prime notation, one of the most common modern notations for differentiation is due to Joseph Louis Lagrange and uses the prime mark, so that the derivative of a function ƒ(x) is denoted ƒ′(x) or simply ƒ′. Similarly, the second and third derivatives are denoted
    (f')'=f'' and (f'')'=f'''.
  • Euler's notation Euler's notation uses a differential operator D, which is applied to a function ƒ to give the first derivative Df. The second derivative is denoted D2ƒ, and the nth derivative is denoted Dnƒ. If y = ƒ(x) is a dependent variable, then often the subscript x is attached to the D to clarify the independent variable x. Euler's notation is then written
    Dx y or Dxf(x).

微分係数の求め方:Rules for finding the derivative

  • Sum rule:
    (af+bg)'=af'+bg' for all functions ƒ and g and all real numbers a and b. 
  • Product rule:
    (fg)'=f'g+fg' for all functions ƒ and g. 
  • Quotient rule:
    (f/g)=(f'g-fg')g^2
  • Chain rule:If f(x) = h(g(x)), then
    f'(x) = h'(g(x))/g'(x).

計算例題:Example computation

f(x)=x^4+sin(x^2)-In(x)e^x+7
is f'(x)=4x^3+cos(x^2)・(2x) -1/x・e^x-In(x)e^x .

Here the second term was computed using the chain rule and third using the product rule. The known derivatives of the elementary functions x2, x4, sin(x), ln(x) and exp(x) = ex, as well as the constant 7, were also used.

d(x^n)/dx=n・x^(n-1)の証明

二項定理を用いる

(xn)' = lim(h→0){(x+h)^n - x^n}/h
=lim(h→0){x^n + nC1hx^(n-1) + nC2hx^(n-2) + ・・・+ nCnh^n - xn}/h
=lim(h→0)(nC1hx^(n-1) + nC2hx^(n-2) + ・・・+ hx^(n-1)
=nC1x^(x-1) = nx^(n-1) ただし、nCk = n!/{k !(n-k !)}である。(証明終)

接線:tangent line

To make the preceding reasoning rigorous, one has to explain what is meant by the difference quotient approaching a certain limiting value k. The precise mathematical formulation was given by Cauchy in the 19th century and is based on the notion of limit. Suppose that the graph does not have a break or a sharp edge at p and it is neither plumb nor too wiggly near p. Then there is a unique value of k such that as h approaches 0, the difference quotient gets closer and closer to k, and the distance between them becomes negligible compared with the size of h, if h is small enough. This leads to the definition of the slope of the tangent line to the graph as the limit of the difference quotients for the function f. This limit is the derivative of the function f at x = a, denoted f ′(a). Using derivatives, the equation of the tangent line can be stated as follows:

tangent line.png

Calculus provides rules for computing the derivatives of functions that are given by formulas, such as the power function, trigonometric functions, exponential function, logarithm, and their various combinations. Thus, equations of the tangents to graphs of all these functions, as well as many others, can be found by the methods of calculus.

偏微分:Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are useful in vector calculus and differential geometry.

The partial derivative of a function f with respect to the variable x is written by various sources as

∂y∂x or ∂f(x)/∂x.

The partial-derivative symbol ∂ is a rounded letter, derived from but distinct from the Greek letter delta. In general, the partial derivative of a function f(x1,...,xn) in the direction xi at the point (a1,...,an) is defined to be:

partialderivative.png

勾配:The gradient

The gradient of f is defined to be the vector field whose components are the partial derivatives of f. That is:

gradiant.png

Here the gradient is written as a row vector, but it is often taken to be a column vector.

gradiant-fig.JPG

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Last-modified: 2010-04-30 (金) 09:52:00 (3483d)