Reciprocal rule
In calculus, the reciprocal rule gives the derivative of the reciprocal of a function "f" in terms of the derivative of "f". The reciprocal rule can be used to show that the power rule holds for negative exponents if it has already been established for positive exponents. Also, one can readily deduce the quotient rule from the reciprocal rule and the product rule. The reciprocal rule states that if "f" is differentiable at a point "x" and "f"("x") ≠ 0 then g("x") = 1/"f"("x") is also differentiable at "x" and formula_1 Proof. This proof relies on the premise  that formula_2 is differentiable at formula_3 and on the theorem that formula_2 is then also necessarily continuous there. Applying the definition of the derivative of formula_5 at formula_6 with formula_7 gives formula_8 The limit of this product exists and is equal to the product of the existing limits of its factors: formula_9 Because of the differentiability of formula_2 at formula_6 the first limit equals formula_12 and because of formula_13 and the continuity of formula_2 at formula_6 the second limit thus yielding formula_16 A weak reciprocal rule that follows algebraically from the product rule. It may be argued that since formula_17  an application of the product rule says that formula_18 and this may be algebraically rearranged to say formula_19 However, this fails to prove that 1/"f" is differentiable at "x"; it is valid only when differentiability of 1/"f" at "x" is already established. In that way, it is a weaker result than the reciprocal rule proved above. However, in the context of differential algebra, in which there is nothing that is not differentiable and in which derivatives are not defined by limits, it is in this way that the reciprocal rule and the more general quotient rule are established. Application to  generalization of the power rule. Often the power rule, stating that formula_20, is proved by methods that are valid only when "n" is a nonnegative integer. This can be extended to negative integers "n" by letting formula_21, where "m" is a positive integer. formula_22 Application to a proof of the quotient rule. The reciprocal rule is a special case of the quotient rule, which states that if "f" and "g" are differentiable at "x" and "g"("x") ≠ 0 then formula_23 The quotient rule can be proved by writing formula_24 and then first applying the product rule, and then applying the  reciprocal rule to the second factor. formula_25 Application to differentiation of trigonometric functions. By using the reciprocal rule one can find the derivative of the secant and cosecant functions. For the secant function: formula_26 The cosecant is treated similarly: formula_27