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Product Rule

The derivative identity

d/(dx)[f(x)g(x)]=lim_(h->0)(f(x+h)g(x+h)-f(x)g(x))/h
(1)
=lim_(h->0)[(f(x+h)g(x+h)-f(x+h)g(x))/h+(f(x+h)g(x)-f(x)g(x))/h]
(2)
=lim_(h->0)[f(x+h)(g(x+h)-g(x))/h+g(x)(f(x+h)-f(x))/h]
(3)
=f(x)g^'(x)+g(x)f^'(x),
(4)

where f^'(x)=df/dx denotes the derivative of f. The Leibniz identity extends the product rule to higher-order derivatives.

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