Lagrange's notation for differentiation

Lagrange's notation for differentiation is the notation for differentiation devised by Joseph Louis Lagrange. Lagrange proposed the notations:

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)

It is done this way as for high numbers of derivatives the number of primes will be come cumbersome to write.

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

and so on.

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

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

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:

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.