# Is taking the derivative inside this integral acceptable?

I have run into an integral where I take the partial derivative of x when the upper limit of the integral is x. This is how I solve it but I am not sure if this is correct. Can someone please check? Thank you!
$$frac{partial}{partial x}int_0^xf(t)dt=int_0^xfrac{partial f(t)}{partial x}dt=0$$

Mathematics Asked on November 18, 2021

No.

Assuming that $$x$$ does not appear free in $$f(t)$$, then by the fundamental theorem of calculus:

$$dfrac{partial ~~}{partial x}int_0^x f(t)mathrm d t=f(x)$$

Answered by Graham Kemp on November 18, 2021

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