by Guest » Wed Nov 13, 2019 8:59 am
"Leibniz's rule" is a generalization of the "Fundamental Theorem of Calculus":
[tex]\frac{d}{dx}\int_{\alpha(x)}^{\beta(x)} f(x, t)dt= \frac{d\beta(x)}{dx}f(x,\beta(x)- \frac{d\alpha(x)}{dx}f(x, \alpha(x))+ \int_{\alpha(x)}^{\beta(x)} \frac{\partial f(x,t)}{\partial x} dt[/tex].
In this problem, [tex]\alpha(x)= 0[/tex], [tex]/beta(x)= x^2[/tex], f is a function of t only so that reduces to [tex]\frac{d}{dx}\int_0^{x^2} f(t)dt= 2x f(x^2)= x cos(\pi x)[/tex].