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  • derivative of inverse function | planetmath.org
    22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error Socket error Could not connect to http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26A24 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26A24 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit Theorem If the real function f f f has an inverse function f subscript f normal f leftarrow and the derivative of f f f at the point x f y x subscript f normal y x f leftarrow y is distinct from zero then f subscript f normal f leftarrow is also differentiable at the point y y y and f y 1 f x superscript subscript f normal normal y 1 superscript f normal x f leftarrow prime y frac 1 f prime x 1 That is the derivatives of a function and its inverse function are inverse numbers of each other provided that they have been taken at the points which correspond to each other it Proof Now we have f f y f x y f subscript f normal y f x y f f leftarrow y f x y The derivatives of both sides must be equal d d y f f y d d y y d d y f subscript f normal y d d y y frac d dy left f f leftarrow y right frac d dy y Using the chain rule we get f f y f y 1 normal superscript f normal subscript f normal y

    Original URL path: http://www.planetmath.org/derivativeofinversefunction (2016-04-25)
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  • derivative of limit function diverges from limit of derivatives | planetmath.org
    cannot always change the order of taking limit and differentiating i e it may well be lim n d d x f n x d d x lim n f n x subscript normal n d d x subscript f n x d d x subscript normal n subscript f n x lim n to infty frac d dx f n x neq frac d dx lim n to infty f n x even in the case that a sequence of continuous and differentiable functions converges uniformly cf Theorem 2 of the parent entry Example The function sequence f n x j 1 n x 3 1 x 2 j x x 1 x 2 n n 1 2 3 fragments subscript f n fragments normal x normal assign superscript subscript j 1 n superscript x 3 superscript 1 superscript x 2 j x x superscript 1 superscript x 2 n italic fragments normal n 1 normal 2 normal 3 normal normal normal displaystyle f n x sum j 1 n frac x 3 1 x 2 j x frac x 1 x 2 n qquad n 1 2 3 ldots 1 provides an instance we consider it on the interval 1 1 1 1 1 1 It s a question of partial sum the converging geometric series x 3 1 x 2 x 3 1 x 2 2 x 3 1 x 2 2 superscript x 3 1 superscript x 2 superscript x 3 superscript 1 superscript x 2 2 superscript x 3 superscript 1 superscript x 2 2 normal frac x 3 1 x 2 frac x 3 1 x 2 2 frac x 3 1 x 2 2 ldots although one cannot use Weierstrass criterion of uniform convergence Since the limit function is f x lim n x x 1 x 2 n x x 1 1 formulae sequence assign f x subscript normal n x x superscript 1 superscript x 2 n x for all x 1 1 f x lim n to infty left x frac x 1 x 2 n right x quad forall x in 1 1 we have sup 1 1 f n x f x sup 1 1 x 1 x 2 n 0 as n formulae sequence subscript supremum 1 1 subscript f n x f x subscript supremum 1 1 x superscript 1 superscript x 2 n normal 0 normal as n sup 1 1 f n x f x sup 1 1 frac x 1 x 2 n longrightarrow 0 quad mbox as n to infty which means by Theorem 1 of the parent entry that the sequence 1 converges uniformly on the interval to the identity function Further the members of the sequence are continuous and differentiable Furthermore f n x 1 1 1 2 n x 2 1 x 2 n 1 superscript subscript f n normal x 1 1 1 2 n superscript x 2 superscript 1 superscript x 2 n 1 f n prime

    Original URL path: http://www.planetmath.org/derivativeoflimitfunctiondivergesfromlimitofderivatives (2016-04-25)
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  • derivative of logarithm with respect to base | planetmath.org
    jcorneli beta sites all modules sparql sparql module User error Socket error Could not connect to http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26A15 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26A15 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit derivative of logarithm with respect to base The formula a log a x ln x ln a 2 a a subscript

    Original URL path: http://www.planetmath.org/derivativeoflogarithmwithrespecttobase (2016-04-25)
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  • derivative of polynomial | planetmath.org
    in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error Socket error Could not connect to http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26A06 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26A06 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit derivative of polynomial Let R R R be an arbitrary commutative ring If f X i 1 n a i X i assign f X superscript subscript i 1 n subscript a i superscript X i f X sum i 1 n a i X i is a polynomial in the ring R X R X R X one can form in a polynomial ring R X Y R X Y R X Y the polynomial f X Y i 1 n a i X Y i f X Y superscript subscript i 1 n subscript a i superscript X Y i f X Y sum i 1 n a i X Y i Expanding this by the powers of Y Y Y yields uniquely the form f X Y f X f 1 X Y f 2 X Y Y 2 assign f X Y f X subscript f 1 X Y subscript f 2 X Y superscript Y 2 displaystyle f X Y f X f 1 X Y f 2 X Y Y 2 1 where f 1 X R X subscript f 1 X R X f 1 X in R X and f 2 X Y R X Y subscript f 2 X Y R X Y f 2

    Original URL path: http://www.planetmath.org/derivativeofpolynomial (2016-04-25)
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  • derivatives of solution of first order ODE | planetmath.org
    3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A12E05 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A12E05 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit derivatives of solution of first order ODE Suppose that f f f is a continuously differentiable function defined on an open subset E E E of ℝ 2 superscript ℝ 2 mathbb R 2 i e it has on E E E the continuous partial derivatives f x x y superscript subscript f x normal x y f x prime x y and f y x y superscript subscript f y normal x y f y prime x y If y x y x y x is a solution of the first order ordinary differential equation d y d x f x y d y d x f x y displaystyle frac dy dx f x y 1 then we have y x f x y x superscript y normal x f x y x displaystyle y prime x f x y x 2 y x f x x y x f y x y x y x superscript y x superscript subscript f x normal x y x superscript subscript f y normal x y x superscript y normal x displaystyle y prime prime x f x prime x y x f y prime x y x y prime x 3 see the general chain rule Thus there exists on E E E the second derivative y x superscript y x y prime prime x which is also continuous More generally we can infer the Theorem If f x y f x y f

    Original URL path: http://www.planetmath.org/derivativesofsolutionoffirstorderode (2016-04-25)
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  • determination of Fourier coefficients | planetmath.org
    skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit determination of Fourier coefficients Suppose that the real function f f f may be presented as sum of the Fourier series f x a 0 2 m 0 a m cos m x b m sin m x f x subscript a 0 2 superscript subscript m 0 subscript a m m x subscript b m m x displaystyle f x frac a 0 2 sum m 0 infty a m cos mx b m sin mx 1 Therefore f f f is periodic with period 2 π 2 π 2 pi For expressing the Fourier coefficients a m subscript a m a m and b m subscript b m b m with the function itself we first multiply the series 1 by cos n x n x cos nx n ℤ n ℤ n in mathbb Z and integrate from π π pi to π π pi Supposing that we can integrate termwise we may write π π f x cos n x d x a 0 2 π π cos n x d x m 0 a m π π cos m x cos n x d x b m π π sin m x cos n x d x superscript subscript π π f x n x d x subscript a 0 2 superscript subscript π π n x d x superscript subscript m 0 subscript a m superscript subscript π π m x n x d x subscript b m superscript subscript π π m x n x d x displaystyle int pi pi f x cos nx dx frac a 0 2 int pi pi cos nx dx sum m 0 infty left a m int pi pi cos mx cos nx dx b m int pi pi sin mx cos nx dx right 2 When n 0 n 0 n 0 the equation 2 reads π π f x d x a 0 2 2 π π a 0 superscript subscript π π f x d x normal subscript a 0 2 2 π π subscript a 0 displaystyle int pi pi f x dx frac a 0 2 cdot 2 pi pi a 0 3 since in the sum of the right hand side only the first addend is distinct from zero When n n n is a positive integer we use the product formulas of the trigonometric identities getting π π cos m x cos n x d x 1 2 π π cos m n x cos m n x d x superscript subscript π π m x n x d x 1 2 superscript subscript π π m n x m n x d x int pi pi cos mx cos nx dx frac 1 2 int

    Original URL path: http://www.planetmath.org/determinationoffouriercoefficients (2016-04-25)
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  • determining envelope | planetmath.org
    0 Fxyc0 F cxyc0 displaystyle begin cases F x y c 0 F prime c x y c 0 end cases 1 I e one may in principle eliminate c c c from such a pair of equations and obtain the equation of an envelope Example 1 Let us determine the envelope of the the family y C x C a 1 C 2 y C x C a 1 superscript C 2 displaystyle y Cx frac Ca sqrt 1 C 2 2 of lines with C C C the parameter a a a is a positive constant Now the pair 1 for the envelope may be written F x y C C x y C a 1 C 2 0 F C x y C x a 1 C 2 1 C 2 0 formulae sequence assign F x y C C x y C a 1 superscript C 2 0 subscript superscript F normal C x y C x a 1 superscript C 2 1 superscript C 2 0 displaystyle F x y C Cx y frac Ca sqrt 1 C 2 0 quad F prime C x y C equiv x frac a 1 C 2 sqrt 1 C 2 0 3 It s easier to first eliminate x x x by taking its expression from the second equation and putting it to the first equation It follows the expression y C 3 a 1 C 2 1 C 2 y superscript C 3 a 1 superscript C 2 1 superscript C 2 y frac C 3 a 1 C 2 sqrt 1 C 2 and so we have the parametric presentation x a 1 C 2 1 C 2 y C 3 a 1 C 2 1 C 2 formulae sequence x a 1 superscript C 2 1 superscript C 2 y superscript C 3 a 1 superscript C 2 1 superscript C 2 x frac a 1 C 2 sqrt 1 C 2 quad y frac C 3 a 1 C 2 sqrt 1 C 2 of the envelope The parameter C C C can be eliminated from these equations by squaring both equations then taking cube roots and adding both equations The result is symmetric equation x 2 3 y 2 3 a 2 3 3 superscript x 2 3 superscript y 2 3 superscript a 2 sqrt 3 x 2 sqrt 3 y 2 sqrt 3 a 2 which represents an astroid But the parametric form tells that the envelope consists only of the left half of the astroid Example 2 What is the envelope of the family y 1 2 a 2 1 4 x a 2 y 1 2 superscript a 2 1 4 superscript x a 2 displaystyle y frac 1 2 a 2 frac 1 4 x a 2 4 of parabolas with a a a the parameter With a fixed a a a the equation presents a parabola which is congruent to the parabola y 1 4 x

    Original URL path: http://www.planetmath.org/determiningenvelope (2016-04-25)
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  • determining integer contraharmonic means | planetmath.org
    dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26B05 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26B05 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error Socket error Could not connect to http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26A24 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26A24 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit For determining effectively values c c c of integer contraharmonic

    Original URL path: http://www.planetmath.org/determiningintegercontraharmonicmeans (2016-04-25)
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