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*微分係数:Derivative [#m4655125]
In calculus, a branch of mathematics, the derivative is a...
The derivative of a function at a chosen input value desc...
*微分と微分係数 [#q082e1b1]
Differentiation is a method to compute the rate at which ...
Differentiation is a method to find an exact value for th...
The idea is to compute the rate of change as the limiting...
In Leibniz's notation, such an infinitesimal change in x ...
dy/dx
suggesting the ratio of two infinitesimal quantities.
The above expression is read as "the derivative of y with...
*導関数:Derivative of the function [#z152d035]
The derivative is the value of the difference quotient as...
#ref(derivative.png)
of the difference quotient as h approaches zero, if this ...
*表記法:notation [#v9170625]
-Leibniz's notation
The notation for derivatives introduced by Gottfried Leib...
dy/dx,df/dx or d/dxf(x).
-Lagrange's notation
Sometimes referred to as prime notation, one of the most ...
(f')'=f'' and (f'')'=f'''.
-Euler's notation
Euler's notation uses a differential operator D, which is...
If y = ƒ(x) is a dependent variable, then often the subsc...
Dx y or Dxf(x).
*微分係数の求め方:Rules for finding the derivative [#w67...
-Sum rule:
(af+bg)'=af'+bg' for all functions ƒ and g and all real ...
-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 [#d30fbf22]
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 an...
*d(x^n)/dx=n・x^(n-1)の証明 [#m62bdfb4]
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...
=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...
*接線:tangent line [#u81a88c8]
To make the preceding reasoning rigorous, one has to expl...
#ref(tangent line.png)
Calculus provides rules for computing the derivatives of ...
*偏微分:Partial derivative [#be00ea85]
In mathematics, a partial derivative of a function of sev...
The partial derivative of a function f with respect to th...
∂y∂x or ∂f(x)/∂x.
The partial-derivative symbol ∂ is a rounded letter, deri...
In general, the partial derivative of a function f(x1,......
#ref(partialderivative.png)
*勾配:The gradient [#g95b9a1a]
The gradient of f is defined to be the vector field whose...
#ref(gradiant.png)
Here the gradient is written as a row vector, but it is o...
#ref(gradiant-fig.JPG)
終了行:
*微分係数:Derivative [#m4655125]
In calculus, a branch of mathematics, the derivative is a...
The derivative of a function at a chosen input value desc...
*微分と微分係数 [#q082e1b1]
Differentiation is a method to compute the rate at which ...
Differentiation is a method to find an exact value for th...
The idea is to compute the rate of change as the limiting...
In Leibniz's notation, such an infinitesimal change in x ...
dy/dx
suggesting the ratio of two infinitesimal quantities.
The above expression is read as "the derivative of y with...
*導関数:Derivative of the function [#z152d035]
The derivative is the value of the difference quotient as...
#ref(derivative.png)
of the difference quotient as h approaches zero, if this ...
*表記法:notation [#v9170625]
-Leibniz's notation
The notation for derivatives introduced by Gottfried Leib...
dy/dx,df/dx or d/dxf(x).
-Lagrange's notation
Sometimes referred to as prime notation, one of the most ...
(f')'=f'' and (f'')'=f'''.
-Euler's notation
Euler's notation uses a differential operator D, which is...
If y = ƒ(x) is a dependent variable, then often the subsc...
Dx y or Dxf(x).
*微分係数の求め方:Rules for finding the derivative [#w67...
-Sum rule:
(af+bg)'=af'+bg' for all functions ƒ and g and all real ...
-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 [#d30fbf22]
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 an...
*d(x^n)/dx=n・x^(n-1)の証明 [#m62bdfb4]
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...
=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...
*接線:tangent line [#u81a88c8]
To make the preceding reasoning rigorous, one has to expl...
#ref(tangent line.png)
Calculus provides rules for computing the derivatives of ...
*偏微分:Partial derivative [#be00ea85]
In mathematics, a partial derivative of a function of sev...
The partial derivative of a function f with respect to th...
∂y∂x or ∂f(x)/∂x.
The partial-derivative symbol ∂ is a rounded letter, deri...
In general, the partial derivative of a function f(x1,......
#ref(partialderivative.png)
*勾配:The gradient [#g95b9a1a]
The gradient of f is defined to be the vector field whose...
#ref(gradiant.png)
Here the gradient is written as a row vector, but it is o...
#ref(gradiant-fig.JPG)
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