Higher Derivatives Higher derivatives are derivatives of derivatives. Given a diﬀerentiable function u = u(x) its derivative u� is a new function, which we may be able to diﬀerentiate again to get (u� )� = u�� . For example, if u(x) = sin x then u� = cos x and u�� = − sin x. We can go on: (u�� )� = u�� = − cos x (u�� = u(3) is called the third derivative of u and u�� is the second derivative) and u�� = u(4) = sin x. The function sin x is a special example – we won’t usually “come back to” the function we started with. Since there’s more than one way to write derivatives, there’s more than one notation for higher derivatives. Notations f � (x) Df f �� (x) D2 f f �� (x) D3 f f (n) (x) Dn f df dx d2 f dx2 d3 f dx3 dn f dxn d f � dx�2 d f dx � �3 d f dx � �n d f dx d The symbols D and represent “operators” which can be applied to a dx function. When you apply one of these operators to a function you get the derivative of that function.

Higher Derivatives

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