Subject: Calculus

Newtons Notation For Differentiation

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 with respect to time denoted as

\frac{dx}{dt}=\dot{x}=x'(t)

Higher Order Derivatives

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;

\frac{d^{2}x}{dt^{2}}=\ddot{x}=x''(t)

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.

Newton's Notation in General

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.
So a function y=f(x) has the following Newton's notation for its first-order derivative with respect to x.

\frac{dy}{dx}=\dot{y}=f'(x)

While its second order derivative is denoted as

\frac{d^{2}y}{dx^{2}}=\ddot{y}=f''(x)

and so on. :-)

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