this was given as an exercise in my first year honours math class. I can't seem to wrap my head around why this is not equal to $xf(t)$. Any help is appreciated! heres the question:
Find the derivative of $F(x) = \int \limits_0^x xf(t) dt$
[Math] Find the derivative of $F(x) = \int_0^x xf(t) dt$
calculusdefinite integralsderivativesintegrationreal-analysis
Best Answer
$x$ is 'constant' with respect to $t$ here - it can take fixed, specific values which have nothing to do with $t$. If it helps, write, say, $$F(3)=\int_{0}^{3}3f(t)dt$$ Now obviously we can factor $3$ out of the integral. Similarly, in general, $$F(x)=x\int_{0}^{x}f(t)dt$$ Now use the product rule and FTC.