If the first derivative, f'(x), of a function, f(x), is itself differentiable, then the result

d/dx f'(x)

is called the second derivative of f(x). Another way of writing this is to use the notation D=d/dx and set

D f(x) = f'(x)
D² f(x) = f''(x)
D³ f(x) = f'''(x) etc. In some cases this process can be continued indefinitely, for example D sin(x) = cos(x)
D ² sin(x) = D cos(x) = -sin(x)
D ³ sin(x) = -cos(x)
etc. Similarly, for polynomials D xⁿ = n x^n-1
D ² xⁿ = n (n-1) x ^n-2
...
D ⁿ xⁿ = n (n-1) (n-2) ... (1) = n!
D ⁿ⁺¹ xⁿ = 0
D ^m x ⁿ = 0 for all m > n

Note: In some cases, the derivative of arbitrary order may be found in close form, but usually it cannot.

Note: Implicit functions may be differentiated repeatedly. The result can be simplified using the fact that the first, second, and higher derivatives can be found in terms of x and y alone.

Last modified Wed Oct 2 21:47:54 CDT 1996