Newton's notation for differentiation

f''(x) for the first derivative

f''(x) for the second derivative

f'''(x) for the third derivative

f⁽ⁿ⁾(x) for the nth derivative (n > 3)

Expressed in terms of Leibniz's notation for differentiation we have:

<math>\frac{d(f(x))}{dx} = f'(x)</math>

<math>\frac{d^2 (f(x))}{dx^2} = f(x)</math>

etc.

Expressed in dot notation we have:

<math>\dot{x} = f'(x) \ \mbox{if} \ x=f(t)</math>

<math>\ddot{x} = f(x)</math>

etc.

Sometimes Newton's notation is more useful than Leibniz's, for example when calculating the derivative at a point.

In Newton's notation, if you know f(x) and you want to calculate f '(x) at a point k, you would write:

<math>f'(k)</math>

and this represents that derivative. For example, if f(x) = x², then f '(3) = 6. The same thing under Leibniz's notation is more cumbersome:

<math>\left. \frac{d(x^2)}{dx} \right|_{x=3} = 6</math>

Leibniz's notation is versatile in that it allows you to specify the variable for differentiation (in the denominator). This is especially relevant for partial differentiation.

Newton did not develop a standard notation[?] for integration, he seemed to use lots of different notations. However the widely adopted notation[?] is Leibniz's notation for integration[?].