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

Mathematics Asked by klein4 on December 1, 2021

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 $f=f(x,t)$.

One 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.

Answered by Jean L. on December 1, 2021

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