DLMF:1.5.E22 (Q145)

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DLMF:1.5.E22
No description defined

    Statements

    Defining formula
    d d x ⁡ ∫ α ⁢ ( x ) β ⁢ ( x ) f ⁢ ( x , y ) ⁢ d y = f ⁢ ( x , β ⁢ ( x ) ) ⁢ β ′ ⁢ ( x ) - f ⁢ ( x , α ⁢ ( x ) ) ⁢ α ′ ⁢ ( x ) + ∫ α ⁢ ( x ) β ⁢ ( x ) ∂ ⁡ f ∂ ⁡ x ⁢ d y . derivative 𝑥 subscript superscript 𝛽 𝑥 𝛼 𝑥 𝑓 𝑥 𝑦 𝑦 𝑓 𝑥 𝛽 𝑥 superscript 𝛽 ′ 𝑥 𝑓 𝑥 𝛼 𝑥 superscript 𝛼 ′ 𝑥 subscript superscript 𝛽 𝑥 𝛼 𝑥 partial-derivative 𝑓 𝑥 𝑦 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\int^{\beta(x)}_{% \alpha(x)}f(x,y)\mathrm{d}y={f(x,\beta(x))\beta^{\prime}(x)-f(x,\alpha(x))% \alpha^{\prime}(x)}+\int^{\beta(x)}_{\alpha(x)}\frac{\partial f}{\partial x}% \mathrm{d}y.}}
    0 references
    DLMF id
    DLMF:1.5.E22
    0 references
    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.m2aadec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    partial derivative of $$f$$ with respect to $$x$$
    Defining formula
    ∂ ⁡ f ∂ ⁡ x partial-derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\partial\NVar{f}}{\partial\NVar{x}}}}
    xml-id
    C1.S5.E3.m4aodec
    0 references
    partial differential of $$x$$
    Defining formula
    ∂ ⁡ x 𝑥 {\displaystyle{\displaystyle\partial\NVar{x}}}
    xml-id
    C1.S5.E3.m2apdec
    0 references
     
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