$\displaystyle \[\frac{d}{{dx}}\left( {\,{x^{1/2}}\int {{x^{3/2}}f(x)dx} } \right)\]$
????
I see how to apply the product rule and I can use the fundamental theorem of calculus on one of the terms. The other term becomes $\displaystyle \frac{1}{2x^{1/2}}\int{x^{3/2}f(x)\,dx$. Can it be simplified from there?
- Hollywood
I have to ask about this one myself.
After the product rule I have:
$\displaystyle \frac{1}{2\sqrt{x}}\int x^{\frac{3}{2}}f(x)dx + \sqrt{x}\frac{d}{dx}\int x^{\frac{3}{2}}f(x)dx$
I don't see how you would differentiate the integral in the second term with the FTOC because it is indefinite, the FTOC is always stated in terms of definite integrals as far as I understand it. I guess if say that $\displaystyle \frac{d}{dx}\int g(x)dx=\frac{d}{dx}(G(x)+C)=g(x)$ is always true so I can say my $\displaystyle x^{\frac{3}{2}}f(x)$ is my $\displaystyle g(x)$ and just say $\displaystyle \frac{d}{dx}\int x^{\frac{3}{2}}f(x)dx= x^{\frac{3}{2}}f(x)$
Or maybe I shouldn't ask if it is true, it is just definition of integral? Where does chain rule come in?