DLMF:1.4.E13 (Q92)

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DLMF:1.4.E13
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    Statements

    Defining formula
    d n d x n ⁡ f ⁢ ( g ⁢ ( x ) ) = ∑ ( n ! m 1 ! ⁢ m 2 ! ⁢ ⋯ ⁢ m n ! ) ⁢ f ( k ) ⁢ ( g ⁢ ( x ) ) ⁢ ( g ′ ⁢ ( x ) 1 ! ) m 1 ⁢ ( g ′′ ⁢ ( x ) 2 ! ) m 2 ⁢ … ⁢ ( g ( n ) ⁢ ( x ) n ! ) m n , derivative 𝑥 𝑛 𝑓 𝑔 𝑥 𝑛 subscript 𝑚 1 subscript 𝑚 2 ⋯ subscript 𝑚 𝑛 superscript 𝑓 𝑘 𝑔 𝑥 superscript superscript 𝑔 ′ 𝑥 1 subscript 𝑚 1 superscript superscript 𝑔 ′′ 𝑥 2 subscript 𝑚 2 … superscript superscript 𝑔 𝑛 𝑥 𝑛 subscript 𝑚 𝑛 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}^{n}}f(g(x))=% \sum\left(\frac{n!}{m_{1}!m_{2}!\cdots m_{n}!}\right)f^{(k)}(g(x))\*\left(% \frac{g^{\prime}(x)}{1!}\right)^{m_{1}}\left(\frac{g^{\prime\prime}(x)}{2!}% \right)^{m_{2}}\dots\left(\frac{g^{(n)}(x)}{n!}\right)^{m_{n}},}}
    0 references
    DLMF id
    DLMF:1.4.E13
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    Symbols used
    derivative of $$f$$ with respect to $$x$$
    Defining formula
    d f d x derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}
    xml-id
    C1.S4.E4.m2acdec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
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    Q10756
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C1.S1.XMD4.m1adec
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    Q10761
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C1.S1.XMD6.m1dec
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