The discussion centers on the justification for differentiating integrals with respect to constants, specifically using the example ∂/∂a[∫e^(− ax^2)dx] = ∫-x^2.e^(-ax^2)dx. The key theorem referenced is Leibniz's rule for differentiating under the integral sign, which allows for this interchange provided the integrand is continuous and continuously differentiable with respect to the variable of differentiation. The conversation clarifies that treating 'a' as a variable enables the integrand to function as a multivariable function, thus validating the differentiation process.
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Regarding your first question: The theorem justifying the interchange of the limit and the integral is usually called something like "Leibniz' rule for differentiating under the integral sign". Assuming the integral limits are fixed (or at least: not functions of the differentiation variable $a$), it is sufficient that the integrand itself is continuous as a function of $x$ and $a$ and continuously differentiable w.r.t. $a$.Yashasvi said:What is the justification for differentiating some integrals with respect to constants in order to obtain result, i.e. ∂/∂a[∫e^(− ax^2).dx] =∫-x^2.e^(-ax^2) dx?I mean what if we say "a" was 3 then differentiating wrt 3 would have no significance?How can we treat it like a multivariable function :/