Quotient rule

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The quotient rule is a fundamental rule of differential calculus . It traces the calculation of the derivative of a quotient of functions back to the calculation of the derivative of the individual functions.

If the functions and are differentiable from an interval D into the real or complex numbers at the point with , then the function f is also with

differentiable at the point and the following applies:


In short:


Quotient rule

The quotient can be interpreted as the slope in a slope triangle whose legs are u (x) and v (x) (see figure). If x increases by Δx, u changes by Δu and v by Δv. The change in slope is then

If one divides by Δx, it follows

If one now forms Limes Δx against 0, then becomes

as claimed.


If you use the short notation , you get, for example, the derivation of the following function:

When multiplied it results

Further derivations

Given After the product rule applies:

According to the reciprocal value rule (results e.g. directly or with the help of the chain rule )


An alternative derivation is only possible with the product rule by deriving the function equation . However, it is implicitly assumed here that it has a derivative at all, that is, that it exists.



The quotient rule for functions is explained in almost every book that deals with differential calculus in a general way. Some specific examples are:

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