If the function one wishes to differentiate, , can be written as
and , then the rule states that the derivative of is
More precisely, if all x in some open set containing the number a satisfy , and and both exist, then exists as well and
And this can be extended to calculate the second derivative as well (you can prove this by taking the derivative of twice). The result of this is:
The derivative of is:
In the example above, the choices
were made. Analogously, the derivative of sin(x)/x2 (when x ≠ 0) is:
- Let some function .
- We wish to find , and this is equivalent to (by the reciprocal rule and the product rule, will be differentiable).
- We also know that .
- By the product rule, we can say that .
- From the equation above,
- Because , the right side simplifies to , which simplifies to
- Hence proved. 
- Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks/Cole. ISBN 0-495-01166-5.
- Larson, Ron; Edwards, Bruce H. (2009). Calculus (9th ed.). Brooks/Cole. ISBN 0-547-16702-4.
- Thomas, George B.; Weir, Maurice D.; Hass, Joel (2010). Thomas' Calculus: Early Transcendentals (12th ed.). Addison-Wesley. ISBN 0-321-58876-2.
- Berresford, Geoffrey; Rockett, Andrew (2008). Brief Applied Calculus (5th ed.). ISBN 0-547-16977-9.