# Quotient rule

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 ${\ displaystyle u (x)}$${\ displaystyle v (x)}$ ${\ displaystyle x = x_ {a}}$${\ displaystyle v (x_ {a}) \ neq 0}$

${\ displaystyle f (x) = {\ frac {u (x)} {v (x)}}}$

differentiable at the point and the following applies: ${\ displaystyle x_ {a}}$

${\ displaystyle f '(x_ {a}) = {\ frac {u' (x_ {a}) \ cdot v (x_ {a}) - u (x_ {a}) \ cdot v '(x_ {a} )} {(v (x_ {a})) ^ {2}}}}$.

In short:

${\ displaystyle \ left ({\ frac {u} {v}} \ right) '= {\ frac {u'v-uv'} {v ^ {2}}}}$

## Derivation

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 ${\ displaystyle u (x) \ over v (x)}$

{\ displaystyle {\ begin {aligned} {\ Delta \ left ({u \ over v} \ right)} & = {{{u + \ Delta u} \ over {v + \ Delta v}} - {u \ over v }} \\ & = {{(u + \ Delta u) \ cdot vu \ cdot (v + \ Delta v)} \ over {(v + \ Delta v) \ cdot v}} \ end {aligned}}}
${\ displaystyle = {{\ Delta u \ cdot vu \ cdot \ Delta v} \ over {v ^ {2} + \ Delta v \ cdot v}}}$

If one divides by Δx, it follows

${\ displaystyle {{\ Delta \ left ({u \ over v} \ right)} \ over {\ Delta x}} = {{{\ Delta u \ over \ Delta x} \ cdot vu \ cdot {\ Delta v \ over \ Delta x}} \ over {v ^ {2} + \ Delta v \ cdot v}}}$

If one now forms Limes Δx against 0, then becomes

${\ displaystyle {\ left ({u \ over v} \ right) '} = {{u' \ cdot vu \ cdot v '} \ over {v ^ {2}}}}$

as claimed.

## example

If you use the short notation , you get, for example, the derivation of the following function: ${\ displaystyle \ left ({\ frac {u} {\ color {Blue} v}} \ right) '= {\ frac {u' \ color {Blue} v \ color {Black} -u \ color {Blue} v '} {\ color {Blue} v \ color {Black} ^ {2}}}}$

{\ displaystyle {\ begin {aligned} f (x) & = {\ frac {x ^ {2} -1} {\ color {Blue} 2-3x}} \\ f '(x) & = {\ frac {2x \ cdot \ color {Blue} (2-3x) \ color {Black} - (x ^ {2} -1) \ cdot \ color {Blue} (- 3)} {(\ color {Blue} 2- 3x \ color {Black}) ^ {2}}} \\\ end {aligned}}}

When multiplied it results ${\ displaystyle f '(x) = {\ frac {-3x ^ {2} + 4x-3} {(2-3x) ^ {2}}}}$

## Further derivations

Given After the product rule applies: ${\ displaystyle f (x) = {\ frac {u (x)} {v (x)}}.}$

{\ displaystyle {\ begin {aligned} f '(x) & = \ left (u (x) \ cdot {\ frac {1} {v (x)}} \ right)' \\ & = u '(x ) {\ frac {1} {v (x)}} + u (x) \ left ({\ frac {1} {v (x)}} \ right) '. \ end {aligned}}}

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

${\ displaystyle \ left ({\ frac {1} {v (x)}} \ right) '= - {\ frac {v' (x)} {v ^ {2} (x)}}.}$

follows:

{\ displaystyle {\ begin {aligned} f '(x) & = u' (x) {\ frac {1} {v (x)}} + u (x) \ left (- {\ frac {v '( x)} {v ^ {2} (x)}} \ right) \\ & = {\ frac {u '(x) v (x) -u (x) v' (x)} {v ^ {2 } (x)}}. \ end {aligned}}}

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. ${\ displaystyle f (x) \ cdot v (x) = u (x)}$${\ displaystyle f (x)}$${\ displaystyle f '(x)}$

${\ displaystyle f '(x) \ cdot v (x) + f (x) \ cdot v' (x) = u '(x)}$

consequently:

{\ displaystyle {\ begin {aligned} f '(x) & = {\ frac {u' (x)} {v (x)}} - {\ frac {u (x)} {v (x)}} \ cdot {\ frac {v '(x)} {v (x)}} \\ & = {\ frac {u' (x) v (x) -u (x) v '(x)} {v ^ {2} (x)}}. \ End {aligned}}}

## literature

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