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## 科学网微分大观园之外文课本

【外文】

1

Calculus(6th   Edition)20

James   Stewart

Brooks   Cole

If we use the traditional notation $y = f ( x )$to indicate that the independent variable is $x$ and the dependent variable is $y$then some common alternative notations for the derivative are as follows

$f ^ { \prime } ( x ) = y ^ { \prime } = \frac { d y } { d x } = \frac { d f } { d x } = \frac { d } { d x } f ( x ) = D f ( x ) = D _ { x } f ( x )$

The symbols  $D$ and  $d/dx$ are called differentiation operators because they indicate the operation of differentiationwhich is the process of calculating a derivative.

The symbol $d y / d x$which was introduced by Leibnizshould not be regarded as a ratio

for the time being）；it is simply a synonym for $\mathrm{f}(\mathrm{x})$.Nonethelessit is a very useful and suggestive notationespecially when used in conjunction with increment notation.Refer-

ring to Equation 3.1.6instantaneous rate of change$= \lim _ { \Delta x \rightarrow 0 } \frac { \Delta y } { \Delta x } = \lim _ { x _ { 2 } - x _ { 1 } } \frac { f \left( x _ { 2 } \right) - f \left( x _ { 1 } \right) } { x _ { 2 } - x _ { 1 } }$），we can rewrite the definition of derivative in Leibniz notation in the form $\frac { d y } { d x } = \lim _ { \Delta x \rightarrow 0 } \frac { \Delta y } { \Delta x }$

If we want to indicate the value of a derivative $d y / d x$ in Leibniz notation at a specific num-ber $a$we use the notation $\left. \frac { d y } { d x } \right| _ { x = a } \quad$ or $\quad \frac { d y } { d x } ] _ { x - a }$

which is a synonym for$f ^ { \prime } ( a )$ .

[1]   James Stewart.Calculus(6th Edition)[M].Brooks Cole.Year,2007:126.

2

Calculus   with Analytic Geometry

George   F.Simmons

The McGraw-Hill Companies

To explain Leibniz's notation,we begin with a function $y=f(x)$ and write the difference quotient $\frac { f ( x + \Delta x ) - f ( x ) } { \Delta x }$

in the form

$\frac { \Delta y } { \Delta x }$

where $\Delta y = f ( x + \Delta x ) - f ( x )$.Here $\Delta y$ is not just any change in$y$it is the specific change that results when the independent variable is changed from $x$ to $x + \Delta x$ .As we knowthe difference quotient $\frac { \Delta y } { \Delta x }$ can be interpreted as the ratio of the change in $y$ to the change in $x$ along the curve $y = f ( x )$and this is the slope of the secantFig2.9.Leibniz wrote the limit of this difference quotientwhich of course is the derivative $f ^ { \prime } ( x )$in the form $d y / d x$read"$dy$ over$d x$".In this notationthe definition of the derivative becomes

$\frac{d y}{d x}=http://blog.sciencenet.cn/\lim _{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}$  (1)

and this is the slope of the tangent in Fig.2.9.Two slightly different equivalent forms of  $dy/dx$ are  $\frac{d f(x)}{d x}$ and $\frac{d}{d x} f(x)$.

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