find $ \frac{d}{dx} \int_{0}^{x} (2x-t)^nf(t) dt$

Question

let $f$ be a continious real value function on $\mathbb{R}$ and $n$ a postive intger .

find $$ \frac{\text{d}}{\text{d}x} \int_{0}^{x} (2x-t)^nf(t) \text{d}t$$

my attempt : im thinking about leibtinitz theorem but i couldnot get the hints...

any hints/solution will be appreciated

Answer 1

From the Leibniz integral rule we get

$\frac{d}{dx} \int_{0}^{x} g(x,t) dt= g(x,x)+\int_0^x g_x(x,t) dt$.

Above we have $g(x,t)=(2x-t)^n f(t)$. Can you proceed ?