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Leibniz's Theorem


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The following result for the n-th derivative of a product:

Theorem

If h(x)=f(x)g(x) for all x, the n-th derivative of h is given by

where the coefficients are binomial coefficients.

For example, to find h(8)(x), when h(x)=x2 sin x, let f(x)=x2 and g(x)=sin x. Then f′(x)=2x and f″(x)=2; and g(8)(x)=sin x, g(7)(x)=−cos x and g(6) (x)=−sin x. So

Subjects: Mathematics.


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