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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 (one 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:
Alternative proof (logarithmic differentiation)
Differentiate both sides,
- 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.