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单词 Leibniz's Theorem
释义

Leibniz's Theorem

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

where the coefficients inline 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, with higher derivatives being zero; and g(8)(x) = sinx, g(7)(x) = −cosx and g(6) (x) = −sinx. So

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