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Lekcja 2 – Obliczanie pochodnych

Jesteś tutaj: Strona główna / Fora / Lekcja 2 – Liczenie pochodnych (VIDEO)

Przeglądasz 15 wpisów - od 1 do 15 (z 25 łącznie)
  • Autor
    Posty
  • #8860
    Krystian Karczyński
    Dyrektor

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    #11196
    Krystian Karczyński
    Dyrektor

    Przykład 28

    bold italic y bold equals bold 2 bold italic a bold italic r bold italic c bold italic c bold italic o bold italic s \fraction numerator square root of bold x over denominator bold 2 end \fraction bold minus square root of bold 3 bold x bold minus bold 2 bold x to the power of bold 2 end root

    y apostrophe equals open parentheses 2 a r c cos \fraction numerator square root of straight x over denominator 2 end \fraction minus square root of 3 straight x minus 2 straight x squared end root close parentheses apostrophe equals open parentheses 2 a r c cos \fraction numerator square root of straight x over denominator 2 end \fraction close parentheses apostrophe minus open parentheses square root of 3 straight x minus 2 straight x squared end root close parentheses apostrophe equals

    equals 2 \times open parentheses a r c cos \fraction numerator square root of straight x over denominator 2 end \fraction close parentheses apostrophe minus \fraction numerator 1 over denominator 2 square root of 3 straight x minus 2 straight x squared end root end \fraction \times open parentheses 3 straight x minus 2 straight x squared close parentheses apostrophe equals
equals 2 \times open parentheses negative \fraction numerator 1 over denominator square root of 1 minus open parentheses \fraction numerator square root of straight x over denominator 2 end \fraction close parentheses squared end root end \fraction close parentheses \times open parentheses \fraction numerator square root of straight x over denominator 2 end \fraction close parentheses apostrophe minus \fraction numerator 1 over denominator 2 square root of 3 straight x minus 2 straight x squared end root end \fraction \times open parentheses 3 \times 1 minus 2 \times 2 straight x close parentheses equals

    equals negative \fraction numerator 2 over denominator square root of 1 minus straight x over 4 end root end \fraction \times 1 half \times open parentheses square root of straight x close parentheses apostrophe minus \fraction numerator 3 minus 4 straight x over denominator 2 square root of 3 straight x minus 2 straight x squared end root end \fraction equals negative \fraction numerator 1 over denominator square root of 4 over 4 minus straight x over 4 end root end \fraction \times \fraction numerator 1 over denominator 2 square root of straight x end \fraction minus \fraction numerator 3 minus 4 straight x over denominator 2 square root of 3 straight x minus 2 straight x squared end root end \fraction equals

    equals negative \fraction numerator 1 over denominator square root of \fraction numerator 4 minus x over denominator 4 end \fraction end root end \fraction \times \fraction numerator 1 over denominator 2 square root of straight x end \fraction minus \fraction numerator 3 minus 4 straight x over denominator 2 square root of 3 straight x minus 2 straight x squared end root end \fraction equals negative \fraction numerator 1 over denominator \fraction numerator square root of 4 minus x end root over denominator square root of 4 end \fraction \times 2 square root of straight x end \fraction minus \fraction numerator 3 minus 4 straight x over denominator 2 square root of 3 straight x minus 2 straight x squared end root end \fraction equals

    equals negative \fraction numerator 1 over denominator \fraction numerator square root of 4 minus x end root over denominator 2 end \fraction \times 2 square root of straight x end \fraction minus \fraction numerator 3 minus 4 straight x over denominator 2 square root of 3 straight x minus 2 straight x squared end root end \fraction equals bold minus \fraction numerator bold 1 over denominator square root of open parentheses bold 4 bold minus bold x close parentheses bold \times bold x end root end \fraction bold minus \fraction numerator bold 3 bold minus bold 4 bold x over denominator bold 2 square root of bold 3 bold x bold minus bold 2 bold x to the power of bold 2 end root end \fraction

    • Ta odpowiedź została zmodyfikowana 2 lata, 4 miesiące temu przez Krystian Karczyński.
    • Ta odpowiedź została zmodyfikowana 1 rok, 11 miesiące temu przez Joanna Grochowska.
    #11195
    Krystian Karczyński
    Dyrektor

    Przykład 29

    y equals l o g subscript 3 open parentheses square root of x squared plus ln x end root close parentheses

    y apostrophe equals open square brackets log subscript 3 open parentheses square root of x squared plus ln x end root close parentheses close square brackets apostrophe equals \fraction numerator 1 over denominator square root of x squared plus ln x end root \times ln 3 end \fraction \times open parentheses square root of x squared plus ln x end root close parentheses apostrophe equals
space space equals \fraction numerator 1 over denominator square root of x squared plus ln x end root \times ln 3 end \fraction \times \fraction numerator 1 over denominator 2 square root of x squared plus ln x end root end \fraction \times open parentheses x squared plus ln x close parentheses apostrophe equals

    equals \fraction numerator 1 over denominator 2 \times open parentheses square root of x squared plus ln x end root close parentheses squared \times ln 3 end \fraction \times open parentheses 2 x plus 1 over x close parentheses equals \fraction numerator bold 2 bold x bold plus bold 1 over bold x over denominator bold 2 open parentheses bold x to the power of bold 2 bold plus bold l bold n bold x close parentheses bold \times bold l bold n bold 3 end \fraction

    • Ta odpowiedź została zmodyfikowana 2 lata, 4 miesiące temu przez Krystian Karczyński.
    • Ta odpowiedź została zmodyfikowana 1 rok, 11 miesiące temu przez Joanna Grochowska.
    #11194
    Krystian Karczyński
    Dyrektor

    Przykład 30

    y equals ln squared open parentheses cos x close parentheses

    y apostrophe equals open square brackets ln squared open parentheses cos x close parentheses close square brackets apostrophe equals open square brackets open parentheses ln open parentheses cos x close parentheses close parentheses squared close square brackets apostrophe equals 2 ln open parentheses cos x close parentheses \times open square brackets ln open parentheses cos x close parentheses close square brackets apostrophe equals
space equals 2 ln open parentheses cos x close parentheses \times \fraction numerator 1 over denominator cos x end \fraction \times open parentheses cos x close parentheses apostrophe equals \fraction numerator 2 ln open parentheses cos x close parentheses over denominator cos x end \fraction \times open parentheses negative sin x close parentheses equals \fraction numerator bold minus bold 2 bold s bold i bold n bold x bold space bold l bold n open parentheses bold c bold o bold s bold x close parentheses over denominator bold c bold o bold s bold x end \fraction

    • Ta odpowiedź została zmodyfikowana 1 rok, 11 miesiące temu przez Joanna Grochowska.
    #11193
    Krystian Karczyński
    Dyrektor

    Przykład 31

    y equals a r c t g cubed open parentheses ln square root of \fraction numerator 1 plus x squared over denominator x end \fraction end root close parentheses

    y apostrophe equals open square brackets a r c t g cubed open parentheses ln square root of \fraction numerator 1 plus x squared over denominator x end \fraction end root close parentheses close square brackets apostrophe space equals space 3 \times a r c t g squared open parentheses ln square root of \fraction numerator 1 plus x squared over denominator x end \fraction end root close parentheses \times open square brackets a r c t g open parentheses ln square root of \fraction numerator 1 plus x squared over denominator x end \fraction end root close parentheses close square brackets apostrophe space equals

    equals space 3 a r c t g squared open parentheses ln square root of \fraction numerator 1 plus x squared over denominator x end \fraction end root close parentheses \times \fraction numerator 1 over denominator open parentheses ln square root of \fraction numerator 1 plus x squared over denominator x end \fraction end root close parentheses squared plus 1 end \fraction \times space open square brackets ln open parentheses square root of \fraction numerator 1 plus x squared over denominator x end \fraction end root close parentheses close square brackets apostrophe space equals
equals \fraction numerator 3 a r c t g squared open parentheses ln square root of \fraction numerator 1 plus x squared over denominator x end \fraction end root close parentheses over denominator ln squared open parentheses square root of \fraction numerator 1 plus x squared over denominator x end \fraction end root close parentheses plus 1 end \fraction \times \fraction numerator 1 over denominator square root of \fraction numerator 1 plus x squared over denominator x end \fraction end root end \fraction space \times open parentheses square root of \fraction numerator 1 plus x squared over denominator x end \fraction end root close parentheses apostrophe space equals

    equals \fraction numerator 3 a r c t g squared open parentheses ln square root of \fraction numerator 1 plus x squared over denominator x end \fraction end root close parentheses over denominator ln squared open parentheses square root of \fraction numerator 1 plus x squared over denominator x end \fraction end root close parentheses plus 1 end \fraction \times \fraction numerator 1 over denominator square root of \fraction numerator 1 plus x squared over denominator x end \fraction end root end \fraction space \times \fraction numerator 1 over denominator 2 square root of \fraction numerator 1 plus x squared over denominator x end \fraction end root end \fraction \times open parentheses \fraction numerator 1 plus x squared over denominator x end \fraction close parentheses apostrophe space equals
equals \fraction numerator 3 a r c t g squared open parentheses ln square root of \fraction numerator 1 plus x squared over denominator x end \fraction end root close parentheses over denominator ln squared open parentheses square root of \fraction numerator 1 plus x squared over denominator x end \fraction end root close parentheses plus 1 end \fraction \times \fraction numerator 1 over denominator 2 open parentheses square root of \fraction numerator 1 plus x squared over denominator x end \fraction end root close parentheses squared end \fraction space \times open parentheses \fraction numerator open parentheses 1 plus x squared close parentheses apostrophe \times x minus open parentheses 1 plus x squared close parentheses \times open parentheses x close parentheses apostrophe over denominator x squared end \fraction close parentheses equals

    equals \fraction numerator 3 a r c t g squared open parentheses ln square root of \fraction numerator 1 plus x squared over denominator x end \fraction end root close parentheses over denominator ln squared open parentheses square root of \fraction numerator 1 plus x squared over denominator x end \fraction end root close parentheses plus 1 end \fraction \times \fraction numerator 1 over denominator 2 \times \fraction numerator 1 plus x squared over denominator x end \fraction end \fraction space \times open parentheses \fraction numerator open parentheses 0 plus 2 x close parentheses \times x minus open parentheses 1 plus x squared close parentheses \times 1 over denominator x squared end \fraction close parentheses equals
equals \fraction numerator 3 a r c t g squared open parentheses ln square root of \fraction numerator 1 plus x squared over denominator x end \fraction end root close parentheses over denominator open parentheses ln squared square root of \fraction numerator 1 plus x squared over denominator x end \fraction end root plus 1 close parentheses end \fraction \times 1 half \times \fraction numerator x over denominator 1 plus x squared end \fraction space \times \fraction numerator 2 x squared minus 1 minus x squared over denominator x squared end \fraction equals
equals \fraction numerator 3 a r c t g squared open parentheses ln square root of \fraction numerator 1 plus x squared over denominator x end \fraction end root close parentheses over denominator 2 \times open parentheses ln squared square root of \fraction numerator 1 plus x squared over denominator x end \fraction end root plus 1 close parentheses end \fraction \times \fraction numerator 1 over denominator 1 plus x squared end \fraction space \times \fraction numerator x squared minus 1 over denominator x end \fraction equals \fraction numerator bold 3 open parentheses bold x to the power of bold 2 bold minus bold 1 close parentheses bold a bold r bold c bold t bold g to the power of bold 2 open parentheses bold ln square root of \fraction numerator bold 1 bold plus bold x to the power of bold 2 over denominator bold x end \fraction end root close parentheses over denominator bold 2 bold x open parentheses bold 1 bold plus bold x to the power of bold 2 close parentheses open parentheses bold ln to the power of bold 2 square root of \fraction numerator bold 1 bold plus bold x to the power of bold 2 over denominator bold x end \fraction end root bold plus bold 1 close parentheses end \fraction

    • Ta odpowiedź została zmodyfikowana 1 rok, 11 miesiące temu przez Joanna Grochowska.
    #11192
    Krystian Karczyński
    Dyrektor

    Przykład 32

    y equals 5 to the power of \fraction numerator x over denominator ln x end \fraction end exponent

    y apostrophe equals open parentheses 5 to the power of \fraction numerator x over denominator ln x end \fraction end exponent close parentheses apostrophe equals 5 to the power of \fraction numerator x over denominator ln x end \fraction end exponent ln 5 \times open parentheses \fraction numerator x over denominator ln x end \fraction close parentheses apostrophe equals 5 to the power of \fraction numerator x over denominator ln x end \fraction end exponent ln 5 \times open parentheses \fraction numerator open parentheses x close parentheses apostrophe \times ln x minus x \times open parentheses ln x close parentheses apostrophe over denominator open parentheses ln x close parentheses squared end \fraction close parentheses equals
equals 5 to the power of \fraction numerator x over denominator ln x end \fraction end exponent ln 5 \times open parentheses \fraction numerator 1 \times ln x minus x \times begin display sty\le 1 over x end sty\le over denominator ln squared x end \fraction close parentheses equals 5 to the power of \fraction numerator x over denominator ln x end \fraction end exponent ln 5 \times \fraction numerator ln x minus 1 over denominator ln squared x end \fraction equals \fraction numerator bold ln bold 5 bold \times bold 5 to the power of \fraction numerator bold x over denominator bold l bold n bold x end \fraction end exponent open parentheses bold ln bold x bold minus bold 1 close parentheses over denominator bold ln to the power of bold 2 bold x end \fraction

    • Ta odpowiedź została zmodyfikowana 2 lata, 4 miesiące temu przez Krystian Karczyński.
    • Ta odpowiedź została zmodyfikowana 1 rok, 11 miesiące temu przez Joanna Grochowska.
    #11191
    Krystian Karczyński
    Dyrektor

    Przykład 33

    y equals x squared 2 to the power of x s i n x

     

    y apostrophe equals open parentheses x squared 2 to the power of x sin x close parentheses apostrophe equals open parentheses x squared \times open parentheses 2 to the power of x s i n x close parentheses close parentheses apostrophe equals open parentheses x squared close parentheses apostrophe \times open parentheses 2 to the power of x sin x close parentheses plus x squared \times open parentheses 2 to the power of x sin x close parentheses apostrophe equals
equals 2 x \times open parentheses 2 to the power of x sin x close parentheses plus x squared \times open square brackets open parentheses 2 to the power of x close parentheses apostrophe sin x plus 2 to the power of x open parentheses sin x close parentheses apostrophe close square brackets equals
equals 2 x open parentheses 2 to the power of x sin x close parentheses plus x squared \times open parentheses 2 to the power of x ln 2 \times sin x plus 2 to the power of x \times cos x close parentheses equals
equals bold 2 bold italic x bold 2 to the power of bold x bold sin bold italic x bold plus bold italic x to the power of bold 2 bold 2 to the power of bold x bold ln bold 2 bold sin bold italic x bold plus bold italic x to the power of bold 2 bold 2 to the power of bold x bold cos bold italic x

    • Ta odpowiedź została zmodyfikowana 2 lata, 4 miesiące temu przez Krystian Karczyński.
    • Ta odpowiedź została zmodyfikowana 1 rok, 11 miesiące temu przez Joanna Grochowska.
    #11190
    Krystian Karczyński
    Dyrektor

    Przykład 34

    y equals ln \fraction numerator 3 sin squared x over denominator e to the power of 4 x end exponent minus cos to the power of 5 x end \fraction

    y apostrophe equals open parentheses ln \fraction numerator 3 sin squared x over denominator e to the power of 4 x end exponent minus cos to the power of 5 x end \fraction close parentheses apostrophe equals \fraction numerator 1 over denominator \fraction numerator 3 sin squared x over denominator e to the power of 4 x end exponent minus cos to the power of 5 x end \fraction end \fraction \times open parentheses \fraction numerator 3 sin squared x over denominator e to the power of 4 x end exponent minus cos to the power of 5 x end \fraction close parentheses apostrophe equals

    equals \fraction numerator e to the power of 4 x end exponent minus cos to the power of 5 x over denominator 3 sin squared x end \fraction \times \fraction numerator open parentheses 3 sin squared x close parentheses apostrophe \times open parentheses e to the power of 4 x end exponent minus cos to the power of 5 x close parentheses minus 3 sin squared x \times open parentheses e to the power of 4 x end exponent minus cos to the power of 5 x close parentheses apostrophe over denominator open parentheses e to the power of 4 x end exponent minus cos to the power of 5 x close parentheses squared end \fraction equals

    equals \fraction numerator e to the power of 4 x end exponent minus cos to the power of 5 x over denominator 3 sin squared x end \fraction \times \fraction numerator 3 \times 2 sin x \times open parentheses sin x close parentheses apostrophe \times open parentheses e to the power of 4 x end exponent minus cos to the power of 5 x close parentheses minus 3 sin squared x \times open parentheses open parentheses e to the power of 4 x end exponent close parentheses apostrophe minus open parentheses cos to the power of 5 x close parentheses apostrophe close parentheses over denominator open parentheses e to the power of 4 x end exponent minus cos to the power of 5 x close parentheses squared end \fraction equals

    equals \fraction numerator e to the power of 4 x end exponent minus cos to the power of 5 x over denominator 3 sin squared x end \fraction \times \fraction numerator 6 sin x \times cos x \times open parentheses e to the power of 4 x end exponent minus cos to the power of 5 x close parentheses minus 3 sin squared x \times open parentheses e to the power of 4 x end exponent \times open parentheses 4 x close parentheses apostrophe minus 5 cos to the power of 4 x \times open parentheses cos x close parentheses apostrophe close parentheses over denominator open parentheses e to the power of 4 x end exponent minus cos to the power of 5 x close parentheses squared end \fraction equals

    equals \fraction numerator e to the power of 4 x end exponent minus cos to the power of 5 x over denominator 3 sin squared x end \fraction \times \fraction numerator 6 sin x cos x open parentheses e to the power of 4 x end exponent minus cos to the power of 5 x close parentheses minus 3 sin squared x \times open parentheses e to the power of 4 x end exponent \times 4 minus 5 cos to the power of 4 x \times open parentheses negative sin x close parentheses close parentheses over denominator open parentheses e to the power of 4 x end exponent minus cos to the power of 5 x close parentheses squared end \fraction equals

    equals \fraction numerator open parentheses bold e to the power of bold 4 bold x end exponent bold minus bold cos to the power of bold 5 bold x close parentheses open square brackets bold 6 bold sin bold x bold cos bold x open parentheses bold e to the power of bold 4 bold x end exponent bold minus bold cos to the power of bold 5 bold x close parentheses bold minus bold 3 bold sin to the power of bold 2 bold x open parentheses bold 4 bold e to the power of bold 4 bold x end exponent bold plus bold 5 bold cos to the power of bold 4 bold x bold sin bold x close parentheses close square brackets over denominator bold 3 bold sin to the power of bold 2 bold x open parentheses bold e to the power of bold 4 bold x end exponent bold minus bold cos to the power of bold 5 bold x close parentheses to the power of bold 2 end \fraction

    equals \fraction numerator open parentheses straight e to the power of 4 straight x end exponent minus cos to the power of 5 straight x close parentheses \times 3 sin x open square brackets 2 cosx open parentheses straight e to the power of 4 straight x end exponent minus cos to the power of 5 straight x close parentheses minus sinx open parentheses 4 straight e to the power of 4 straight x end exponent plus 5 cos to the power of 4 xsinx close parentheses close square brackets over denominator 3 sin squared straight x open parentheses straight e to the power of 4 straight x end exponent minus cos to the power of 5 straight x close parentheses squared end \fraction equals

    equals \fraction numerator 2 cosx open parentheses straight e to the power of 4 straight x end exponent minus cos to the power of 5 straight x close parentheses minus sinx open parentheses 4 straight e to the power of 4 straight x end exponent plus 5 cos to the power of 4 xsinx close parentheses over denominator sinx open parentheses straight e to the power of 4 straight x end exponent minus cos to the power of 5 straight x close parentheses end \fraction

    • Ta odpowiedź została zmodyfikowana 1 rok, 11 miesiące temu przez Joanna Grochowska.
    #11189
    Anonim
    Nieaktywne

    Super. Dzięki wielkie. 🙂

    Fajnie jest wiedzieć, że można liczyć na pomoc w przypadku niejasności z zadaniami domowymi.

    #11184
    Krystian Karczyński
    Dyrektor

    Przykład 35

    y equals x to the power of x squared end exponent

    y apostrophe equals open parentheses x to the power of x squared end exponent close parentheses apostrophe equals open square brackets x to the power of y equals e to the power of ln space x to the power of y end exponent equals e to the power of y ln x end exponent close square brackets equals open parentheses e to the power of x squared ln x end exponent close parentheses apostrophe equals e to the power of x squared ln x end exponent \times open parentheses x squared ln x close parentheses apostrophe equals
equals e to the power of x squared ln x end exponent \times open parentheses open parentheses x squared close parentheses apostrophe \times ln x plus x squared open parentheses ln x close parentheses apostrophe close parentheses equals e to the power of x squared ln x end exponent \times open parentheses 2 x \times ln x plus x squared \times 1 over x close parentheses equals bold italic e to the power of bold x to the power of bold 2 bold ln bold x end exponent bold \times open parentheses bold 2 bold x bold ln bold x bold plus bold x close parentheses

    • Ta odpowiedź została zmodyfikowana 1 rok, 11 miesiące temu przez Joanna Grochowska.
    #11183
    Krystian Karczyński
    Dyrektor

    Przykład 36

    y equals open parentheses 1 plus x close parentheses to the power of s i n x end exponent

    Analogicznie jak przykład wyżej, „(funkcja)^(funkcja)” przekształcam do wyrażenia z logarytmem naturalnym i liczbą e.

    y apostrophe equals open parentheses open parentheses 1 plus x close parentheses to the power of sin x end exponent close parentheses apostrophe equals open parentheses e to the power of sin x \times ln open parentheses 1 plus x close parentheses end exponent close parentheses apostrophe equals e to the power of sin x \times ln open parentheses 1 plus x close parentheses end exponent \times open parentheses sin x \times ln open parentheses 1 plus x close parentheses close parentheses apostrophe equals

    equals e to the power of sin x \times ln open parentheses 1 plus x close parentheses end exponent \times open parentheses open parentheses sin x close parentheses apostrophe \times ln open parentheses 1 plus x close parentheses plus sin x \times open parentheses ln open parentheses 1 plus x close parentheses close parentheses apostrophe close parentheses equals

    equals e to the power of sin x \times ln open parentheses 1 plus x close parentheses end exponent \times open parentheses cos x \times ln open parentheses 1 plus x close parentheses plus sin x \times \fraction numerator 1 over denominator 1 plus x end \fraction \times open parentheses 1 plus x close parentheses apostrophe close parentheses equals

    equals e to the power of sin x \times ln open parentheses 1 plus x close parentheses end exponent open parentheses cos x ln open parentheses 1 plus x close parentheses plus \fraction numerator sin x over denominator 1 plus x end \fraction \times open parentheses 0 plus 1 close parentheses close parentheses equals

    bold equals bold italic e to the power of bold s bold i bold n bold x bold \times bold l bold n open parentheses bold 1 bold plus bold x close parentheses end exponent open parentheses bold c bold o bold s bold x bold l bold n open parentheses bold 1 bold plus bold x close parentheses bold plus \fraction numerator bold s bold i bold n bold x over denominator bold 1 bold plus bold x end \fraction close parentheses

    • Ta odpowiedź została zmodyfikowana 1 rok, 11 miesiące temu przez Joanna Grochowska.
    #11182
    Krystian Karczyński
    Dyrektor

    Przykład 37

    y equals open parentheses ln x close parentheses to the power of x

    y apostrophe equals open parentheses open parentheses ln x close parentheses to the power of x close parentheses apostrophe equals open parentheses e to the power of x \times ln open parentheses ln x close parentheses end exponent close parentheses apostrophe equals e to the power of x \times ln open parentheses ln x close parentheses end exponent \times open parentheses x \times ln open parentheses ln x close parentheses close parentheses apostrophe equals

    equals e to the power of x \times ln open parentheses ln x close parentheses end exponent \times open parentheses open parentheses x close parentheses apostrophe \times ln open parentheses ln x close parentheses plus x \times open square brackets ln open parentheses ln x close parentheses close square brackets apostrophe close parentheses equals

    equals e to the power of x \times ln open parentheses ln x close parentheses end exponent \times open parentheses 1 \times ln open parentheses ln x close parentheses plus x \times \fraction numerator 1 over denominator ln x end \fraction \times open parentheses ln x close parentheses apostrophe close parentheses equals space

    equals e to the power of x \times ln open parentheses ln x close parentheses end exponent open parentheses ln open parentheses ln x close parentheses plus x \times \fraction numerator 1 over denominator ln x end \fraction \times 1 over x close parentheses equals bold space bold italic e to the power of bold x bold \times bold l bold n open parentheses bold l bold n bold x close parentheses end exponent open parentheses bold l bold n open parentheses bold l bold n bold x close parentheses bold plus \fraction numerator bold 1 over denominator bold l bold n bold x end \fraction close parentheses

    • Ta odpowiedź została zmodyfikowana 1 rok, 11 miesiące temu przez Joanna Grochowska.
    #10904
    Krystian Karczyński
    Dyrektor

    Przykład 20

    y equals ln cube root of sin x end root

    y apostrophe equals open parentheses ln cube root of sin x end root close parentheses apostrophe equals \fraction numerator 1 over denominator cube root of sin x end root end \fraction \times open parentheses cube root of sin x end root close parentheses apostrophe equals \fraction numerator 1 over denominator cube root of sin x end root end \fraction \times open parentheses open parentheses sin x close parentheses to the power of 1 third end exponent close parentheses apostrophe equals
space space equals \fraction numerator 1 over denominator cube root of sin x end root end \fraction \times 1 third open parentheses sin x close parentheses to the power of 1 third minus 1 end exponent \times open parentheses sin x close parentheses apostrophe equals \fraction numerator open parentheses bold s bold i bold n bold x close parentheses to the power of bold minus bold 2 over bold 3 end exponent bold space bold c bold o bold s bold x over denominator bold 3 bold space root index bold 3 of bold s bold i bold n bold x end root end \fraction

    • Ta odpowiedź została zmodyfikowana 1 rok, 11 miesiące temu przez Joanna Grochowska.
    #10888
    Tomasz Duda Duda
    Student

    poproszę 27.

    #10882
    Krystian Karczyński
    Dyrektor

    Przykład 27

    y equals 4 minus a r c t g \fraction numerator 2 x over denominator 1 plus x end \fraction

    y apostrophe equals open parentheses 4 minus a r c t g \fraction numerator 2 x over denominator 1 plus x end \fraction close parentheses apostrophe equals open parentheses 4 close parentheses apostrophe minus open parentheses a r c t g \fraction numerator 2 x over denominator 1 plus x end \fraction close parentheses apostrophe equals 0 minus \fraction numerator 1 over denominator open parentheses \fraction numerator 2 x over denominator 1 plus x end \fraction close parentheses squared plus 1 end \fraction \times open parentheses \fraction numerator 2 x over denominator 1 plus x end \fraction close parentheses apostrophe equals
equals negative \fraction numerator 1 over denominator \fraction numerator 4 x squared over denominator open parentheses 1 plus x close parentheses squared end \fraction plus 1 end \fraction \times open parentheses \fraction numerator open parentheses 2 x close parentheses apostrophe \times open parentheses 1 plus x close parentheses minus 2 x \times open parentheses 1 plus x close parentheses apostrophe over denominator open parentheses 1 plus x close parentheses squared end \fraction close parentheses equals
equals negative \fraction numerator 1 over denominator \fraction numerator 4 x squared over denominator open parentheses 1 plus x close parentheses squared end \fraction plus open parentheses 1 plus x close parentheses squared over open parentheses 1 plus x close parentheses squared end \fraction \times open parentheses \fraction numerator 2 \times open parentheses 1 plus x close parentheses minus 2 x \times open parentheses 0 plus 1 close parentheses over denominator open parentheses 1 plus x close parentheses squared end \fraction close parentheses equals negative \fraction numerator 1 over denominator \fraction numerator 4 x squared plus open parentheses 1 plus x close parentheses squared over denominator open parentheses 1 plus x close parentheses squared end \fraction end \fraction \times \fraction numerator 2 plus 2 x minus 2 x over denominator open parentheses 1 plus x close parentheses squared end \fraction equals
equals negative \fraction numerator open parentheses 1 plus x close parentheses squared over denominator 4 x squared plus open parentheses 1 plus x close parentheses squared end \fraction \times 2 over open parentheses 1 plus x close parentheses squared equals negative \fraction numerator 1 over denominator 4 x squared plus 1 plus 2 x plus x squared end \fraction \times 2 over 1 equals bold minus \fraction numerator bold 2 over denominator bold 5 bold x to the power of bold 2 bold plus bold 2 bold x bold plus bold 1 end \fraction

    • Ta odpowiedź została zmodyfikowana 1 rok, 11 miesiące temu przez Joanna Grochowska.
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