MHB Differentiating wrt constant to evaluate integral

[Yashasvi Grover]

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 :/

 

[S.G. Janssens]

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 :/

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. $a$.

Regarding your second question: Nothing stops us from regarding $a$ as a variable (so the integrand indeed becomes a function of two variables). Also, differentiation does not commute with evaluation.