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Higher Derivatives - Definition


The second derivative of a real function f is the derivative of the derivative of f, and is denoted by f". The third derivative off is the derivative of the second derivative, and is denoted by f''', or f(3). In general, the nth derivative of f is denoted by f(n).

If y depends on x, y = f (x), then the second differential of y is defined to be

d2y = f"(x)dx2.

In general the nth differential of y is defined by

dny = f(n)(x) dxn.

Here dx2 means (dx)2 and dxn means (dx)n.

We thus have the alternative notations


for the second and nth derivatives. The notation

y" = f "M,     y(n) = f(n)(x)

is also used.

Last Update: 2006-11-25