How do you take the derivative $frac{d}{dx} int_a^x f(x,t) dt$?

Question

How does one take the derivative for the function $g(x) = int_a^xf(x,t) dt$?
$$ frac{d}{dx}g(x) = frac{d}{dx}int_a^xf(x,t) dt $$
For example, how would one find $frac{d}{dx}int_0^x x + t  dt$?
If $f=f(t)$, then I know I can just use the fundamental theorem, but here $f=f(x,t)$.

Jean L. · Answer

Hint:
Define $phi(x,y)=int^x_af(y,t),dt$ and $g(x)=(x,x)$. Then your function is $phicirc g$ and you can apply the chain rule.

How do you take the derivative $frac{d}{dx} int_a^x f(x,t) dt$?

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