Math 2.0

Real functions Proposition

Reciprocal rule

Builds on DerivativeDifferentiable functions are continuous

Statement

Let gg be a real function on an open interval II, differentiable at a∈Ia \in I, with g(a)≠0g(a) \ne 0. Then 1/g1/g is differentiable at aa, and (1g)′(a)=−g′(a)g(a)2.\left( \frac{1}{g} \right)'(a) = -\frac{g'(a)}{g(a)^2}.

Remarks

The difference quotient is −g(a+h)−g(a)h⋅1g(a+h)g(a)-\frac{g(a + h) - g(a)}{h} \cdot \frac{1}{g(a + h) g(a)}. Continuity of gg at aa gives the limit. For example, (1/x)′=−1/x2(1/x)' = -1/x^2 for x≠0x \ne 0. Mathlib states it as HasDerivAt.inv.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph
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…statementReciprocal rule, statement, no evidence levelReciprocal 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.
  • no evidence level
  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. Reciprocal rule statement

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Reciprocal_rule, Proof
About this page
Tag0QD4E5JYPT
URIm2://0QD4E5JYPT
Kindstatement, proposition
WritingW0 stub
MSC26A24
StatusThis copy of the site has no build data. Run m2 status 0QD4E5JYPT.
UsesDerivative, Differentiable functions are continuous
Same as

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by