let-considere-f-and-u-differenciable-function-prove-that-d-dt-a-u-t-f-t-x-dx-a-u-t-f-t-t-x-dx-f-t-u-t-u-t-

Tinku Tara June 4, 2023 Differentiation 0 Comments

Question Number 33122 by abdo imad last updated on 10/Apr/18

let considere f and u differenciable function prove that (d/dt)( ∫_a ^(u(t)) f(t,x)dx)=∫_a ^(u(t)) (∂f/∂t)(t,x)dx +f(t,u(t))u^′ (t)

$${let}\:{considere}\:{f}\:{and}\:{u}\:{differenciable}\:{function}\:{prove} \\ $$$${that}\:\frac{{d}}{{dt}}\left(\:\int_{{a}} ^{{u}\left({t}\right)} {f}\left({t},{x}\right){dx}\right)=\int_{{a}} ^{{u}\left({t}\right)} \:\frac{\partial{f}}{\partial{t}}\left({t},{x}\right){dx}\:+{f}\left({t},{u}\left({t}\right)\right){u}^{'} \left({t}\right) \\ $$

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