Leibniz integral rule

It gives the derivative with respect to x of an integral whose limits and integrand both depend on x. It is used to differentiate under the integral sign when the limits of integration are functions of the parameter.

\frac{d}{d x} \int_{α (x)}^{β (x)} f (x, y) d y = f (x, β (x)) β^{'} (x) - f (x, α (x)) α^{'} (x) + \int_{α (x)}^{β (x)} \frac{\partial f ( x , y )}{\partial x} d y

\frac{\mathrm{d}}{\mathrm{d}x} \int_{\alpha(x)}^{\beta(x)} f(x, y) \, \mathrm{d}y = f(x, \beta(x)) \beta'(x) - f(x, \alpha(x)) \alpha'(x) + \int_{\alpha(x)}^{\beta(x)} \frac{\partial f(x, y)}{\partial x} \, \mathrm{d}y

What the symbols mean

$x$	the parameter with respect to which the derivative is taken
$y$	the variable of integration
$f (x, y)$	the integrand, a function of both x and y
$α (x)$	the lower limit of integration, a differentiable function of x
$β (x)$	the upper limit of integration, a differentiable function of x
$α^{'} (x)$	the derivative of the lower limit with respect to x
$β^{'} (x)$	the derivative of the upper limit with respect to x
$\frac{\partial f ( x , y )}{\partial x}$	the partial derivative of the integrand with respect to x

About this formula

FieldMathematics
TopicCalculus
LevelUniversity
Also calledLeibniz rule for differentiating under the integral sign, Leibniz's theorem for differentiation of integrals, differentiation under the integral sign, derivative of an integral with variable limits, Leibniz integral formula, differentiating an integral with respect to a parameter
SourceWikidata Q2996637

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What the symbols mean

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