Question

use the definition of the derivative to evaluate v=4/2pir^3

Answer 1

$f'(x)=\displaystyle{\lim_{h\to 0}}\frac{f(x+h)-f(x)}{h}$

$\frac{dv}{dr}=\frac{4\pi}{2}\displaystyle{\lim_{h\to 0}}\frac{v(r+h)-f(r)}{h}=2\pi \displaystyle{\lim_{h\to 0}}\frac{(r+h)^3-r^3}{h}=$

$=2\pi \displaystyle{\lim_{h\to 0}}\frac{r^3+3r^2h+3rh^2+h^3-r^3}{h}=2\pi \displaystyle{\lim_{h\to 0}}(3r^2+3rh+h^2)=$

$=2\pi\cdot 3r^2=6\pi r^2$