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## Quick Reference

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*)=*x*^{2} sin *x*, let *f*(*x*)=*x*^{2} and *g*(*x*)=sin *x*. Then *f*′(*x*)=2*x* 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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