Math 2.0

Real functions Differentiation Theorem

Quotient rule

Builds on Product ruleReciprocal rule

Statement

Let ff and gg be real functions on an open interval II, both differentiable at a∈Ia \in I, with g(a)≠0g(a) \ne 0. Then f/gf/g is differentiable at aa, and (fg)′(a)=f′(a)g(a)−f(a)g′(a)g(a)2.\left( \frac{f}{g} \right)'(a) = \frac{f'(a) g(a) - f(a) g'(a)}{g(a)^2}.

Remarks

Write f/g=f⋅(1/g)f/g = f \cdot (1/g), and use the product rule and the reciprocal rule. For example, it gives (tan⁡x)′=1/cos⁡2x(\tan x)' = 1/\cos^2 x. Mathlib states it as HasDerivAt.div.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionDifferentiable function, definition, no evidence levelDifferentiable functi…definitionDifferentiable functions are continuous, statement, no evidence levelDifferentiable functi…statementProduct rule, statement, no evidence levelProduct rulestatementReciprocal rule, statement, no evidence levelReciprocal rulestatementQuotient rule, statement, no evidence levelQuotient rulestatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Limit point of a set definition
  2. Limit of a function definition
  3. Continuity at a point definition
  4. Derivative definition
  5. Differentiable function definition
  6. Differentiable functions are continuous statement
  7. Product rule statement
  8. Reciprocal rule statement
  9. Quotient rule statement

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Quotient_rule, lead section
About this page
Tag0BS0NJPKVE
URIm2://0BS0NJPKVE
Kindstatement, theorem
WritingW1 stated
MSC26A24
StatusThis copy of the site has no build data. Run m2 status 0BS0NJPKVE.
UsesProduct rule, Reciprocal rule
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