Product Rule

The Derivative identity

${d\over dx} [f(x)g(x)] = \lim_{h\to 0} {f(x+h)g(x+h)-f(x)g(x)\over h}$

$\quad = \lim_{h\to 0} \left[{f(x+h)g(x+h)-f(x+h)g(x)\over h}+{f(x+h)g(x)-f(x)g(x)\over h}\right]$

$\quad = \lim_{h\to 0} \left[{f(x+h) {g(x+h)-g(x)\over h}+ g(x) {f(x+h)-f(x)\over h}}\right]= f(x)g'(x)+g(x)f'(x).$

See also Chain Rule, Exponent Laws, Quotient Rule

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972.

${d\over dx} [f(x)g(x)] = \lim_{h\to 0} {f(x+h)g(x+h)-f(x)g(x)\over h}$
$\quad = \lim_{h\to 0} \left[{f(x+h)g(x+h)-f(x+h)g(x)\over h}+{f(x+h)g(x)-f(x)g(x)\over h}\right]$
$\quad = \lim_{h\to 0} \left[{f(x+h) {g(x+h)-g(x)\over h}+ g(x) {f(x+h)-f(x)\over h}}\right]= f(x)g'(x)+g(x)f'(x).$