Subject: Calculus

Newtons Notation For Differentiation

Calculus.NewtonsNotationForDifferentiation History

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July 13, 2011 by matthew_suan -
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July 13, 2011 by matthew_suan -
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Sir Isaac Newton independently discovered calculus and used it in describing his laws of motion and gravitation. His notation for differentiation is also called "dot notation" wherein he used a dot placed over a function's name to denote the derivative of that function. Usually, Newton's use of dot notation pertains only to mechanics wherein it mostly tackles the derivative of a function with respect to time. For example, a function {$x=f(t)$} has its first derivative derivative with respect to time denoted as
to:
Sir Isaac Newton independently discovered [[calculus]] and used it in describing his laws of motion and gravitation. His [[notations|notation]] for [[differential calculus|differentiation]] is also called "dot notation" wherein he used a dot placed over a function's name to denote the derivative of that function. Usually, Newton's use of dot notation pertains only to mechanics wherein it mostly tackles the derivative of a function with respect to time. For example, a function {$x=f(t)$} has its first derivative with respect to time denoted as
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While the first order derivative notation involves a single dot at the top of the function, Newton intuitively placed two dots at the top of the function to denote the second order derivative of a function. Thus, the second order derivative of the function above with respect to time has the following notation;
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While the first order [[derivative]] notation involves a single dot at the top of the function, Newton intuitively placed two dots at the top of the function to denote the second order derivative of a function. Thus, the second order derivative of the function above with respect to time has the following notation;
June 26, 2011 by matthew_suan -
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!!!!NEXT!
[[Leibniz's notation for differentiation]]
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'''NEXT TOPIC''': [[Leibniz's notation for differentiation]]
November 10, 2010 by matthew_suan -
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!!!!NEXT!
[[Leibniz's notation for differentiation]]
November 07, 2010 by matthew_suan -
Changed lines 2-3 from:
Sir Isaac Newton independently discovered calculus and used it in describing his laws of motion and gravitation. His notation for differentiation is also called "dot notation" wherein he used a dot placed over a function name to denote the derivative of that function. Usually, Newton's use of dot notation pertains only to mechanics wherein it mostly tackles on the derivative of a function with respect to time. For example, a function {$x=f(t)$} has its first derivative derivative with respect to time denoted as
to:
Sir Isaac Newton independently discovered calculus and used it in describing his laws of motion and gravitation. His notation for differentiation is also called "dot notation" wherein he used a dot placed over a function's name to denote the derivative of that function. Usually, Newton's use of dot notation pertains only to mechanics wherein it mostly tackles the derivative of a function with respect to time. For example, a function {$x=f(t)$} has its first derivative derivative with respect to time denoted as
Changed lines 10-11 from:
While the first order derivative notation involves a single dot at the top of the function, Newton intuitively place two dots at the top of the function to denote the second order derivative of a function. Thus, the second order derivative of the function above with respect to time has the following notation;
to:
While the first order derivative notation involves a single dot at the top of the function, Newton intuitively placed two dots at the top of the function to denote the second order derivative of a function. Thus, the second order derivative of the function above with respect to time has the following notation;
Changed lines 14-15 from:
Also, third order derivatives utilizes three dots at the top of the function and so on up the highest order but Newton neglects derivatives at this order due to its insignificant application in mechanics and physics. Unlike in the first order derivative where the observable being measured is the velocity and the acceleration for the second order, higher order derivatives has almost no physical meaning at all and thus seldom being studied.
to:
Also, third order derivatives utilizes three dots at the top of the function and so on up the highest order but Newton neglects derivatives at this order due to its insignificant application in mechanics and physics. Unlike in the first order derivative where the item being measured is the velocity and the acceleration for the second order, higher order derivatives have almost no physical meaning at all and thus are seldom studied.
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As we saw earlier, Newton's derivatives only involves functions and its derivatives with respect to time. So does it mean that a time-independent function has no corresponding Newton's notation for its derivatives? Well, the "dot notation" as devised by Newton is commonly attributed to derivatives with respect to time such as velocity and acceleration but it does not limit to time-dependent functions alone. Although Newton used this notation to derivatives with respect to time, it does not prohibit us from borrowing his devise and use it to denote the derivative of a time-independent functions as well.\\
to:
As we saw earlier, Newton's derivatives only involve functions and their derivatives with respect to time. So does it mean that a time-independent function has no corresponding Newton's notation for its derivatives? Well, the "dot notation" as devised by Newton is commonly attributed to derivatives with respect to time such as velocity and acceleration but it does not limit to time-dependent functions alone. Although Newton used this notation to derivatives with respect to time, it does not prohibit us from borrowing his devise and using it to denote the derivative of a time-independent functions as well.\\
November 06, 2010 by matthew_suan -
November 06, 2010 by matthew_suan -
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%cframe align=center%'+{$\frac{d^{2}y}{dx^{2}}=\ddot{y}=f''(t)$}+'
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%cframe align=center%'+{$\frac{d^{2}y}{dx^{2}}=\ddot{y}=f''(x)$}+'
November 06, 2010 by matthew_suan -
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(:table border=0 cellpadding=3 cellspacing=0 align=center:)
(:cellnr:)
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(:tableend:)
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!!!Newton's Notation in General
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!!Newton's Notation in General
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%cframe align=center%'+{$\frac{d^{2}x}{dt^{2}}=\ddot{x}=x''(t)$}+'
to:
%cframe align=center%'+{$\frac{dy}{dx}=\dot{y}=f'(x)$}+'

While its second order derivative is denoted as

%cframe align=center%'+{$\frac{d^{2}y}{dx^{2}}=\ddot{y}=f''(t)$}+'

and so on. :-)\\
November 06, 2010 by matthew_suan -
Changed lines 2-3 from:
Sir Isaac Newton independently discovered calculus and use it in describing his laws of motion and gravitation. His notation for differentiation is also called "dot notation" wherein he used a dot placed over a function name to denote the derivative of that function. Usually, Newton's use of dot notation pertains only to mechanics wherein it mostly tackles on the derivative of a function with respect to time. For example, a function {$x=f(t)$} has its first derivative derivative with respect to time denoted as
to:
Sir Isaac Newton independently discovered calculus and used it in describing his laws of motion and gravitation. His notation for differentiation is also called "dot notation" wherein he used a dot placed over a function name to denote the derivative of that function. Usually, Newton's use of dot notation pertains only to mechanics wherein it mostly tackles on the derivative of a function with respect to time. For example, a function {$x=f(t)$} has its first derivative derivative with respect to time denoted as
Changed lines 17-19 from:
Also, third order derivatives utilizes three dots at the top of the function and so on up the highest order but Newton neglects derivatives at this order due to its insignificant application in mechanics and physics.

!!!General
to:
Also, third order derivatives utilizes three dots at the top of the function and so on up the highest order but Newton neglects derivatives at this order due to its insignificant application in mechanics and physics. Unlike in the first order derivative where the observable being measured is the velocity and the acceleration for the second order, higher order derivatives has almost no physical meaning at all and thus seldom being studied.

!!!Newton's Notation in General
As we saw earlier, Newton's derivatives only involves functions and its derivatives with respect to time. So does it mean that a time-independent function has no corresponding Newton's notation for its derivatives? Well, the "dot notation" as devised by Newton is commonly attributed to derivatives with respect to time such as velocity and acceleration but it does not limit to time-dependent functions alone. Although Newton used this notation to derivatives with respect to time, it does not prohibit us from borrowing his devise and use it to denote the derivative of a time-independent functions as well.\\
So a function {$y=f(x)$} has the following Newton's notation for its first-order derivative with respect to {$x$}.


%cframe align=center%'+{$\frac{d^{2}x}{dt^{2}}=\ddot{x}=x''(t)$}+'
November 06, 2010 by matthew_suan -
Changed lines 14-15 from:
%cframe align=center%'+{$\frac{dx}{dt}=\dot{x}=x'(t)$}+'
(:tableend:)
to:
%cframe align=center%'+{$\frac{d^{2}x}{dt^{2}}=\ddot{x}=x''(t)$}+'
(:tableend:)

Also, third order derivatives utilizes three dots at the top of the function and so on up the highest order but Newton neglects derivatives at this order due to its insignificant application in mechanics and physics.

!!!General
November 06, 2010 by matthew_suan -
Changed lines 1-15 from:
Sir Isaac Newton independently discovered calculus and use it in describing his laws of motion and gravitation. His notation for differentiation is also called "dot notation"
to:
%lframe%Attach:newton.jpg
Sir Isaac Newton independently discovered calculus and use it in describing his laws of motion and gravitation. His notation for differentiation is also called "dot notation" wherein he used a dot placed over a function name to denote the derivative of that function. Usually, Newton's use of dot notation pertains only to mechanics wherein it mostly tackles on the derivative of a function with respect to time. For example, a function {$x=f(t)$} has its first derivative derivative with respect to time denoted as

(:table border=0 cellpadding=3 cellspacing=0 align=center:)
(:cellnr:)
%cframe align=center%'+{$\frac{dx}{dt}=\dot{x}=x'(t)$}+'
(:tableend:)

!!Higher Order Derivatives
While the first order derivative notation involves a single dot at the top of the function, Newton intuitively place two dots at the top of the function to denote the second order derivative of a function. Thus, the second order derivative of the function above with respect to time has the following notation;

(:table border=0 cellpadding=3 cellspacing=0 align=center:)
(:cellnr:)
%cframe align=center%'+{$\frac{dx}{dt}=\dot{x}=x'(t)$}+'
(:tableend:)
November 06, 2010 by matthew_suan -
Added line 1:
Sir Isaac Newton independently discovered calculus and use it in describing his laws of motion and gravitation. His notation for differentiation is also called "dot notation"