Question

Use the definition of the derivative to differentiate V=(4)/(2)\pi r^(3).

Answer 1

Rewriting the expression given

$V=\frac{4}{2}\pi\>r^3$

$V=2\pi\>r^3$

Derivative is defined as $\frac{\Delta\>y}{\Delta\>x}$

In a graph of V against $r$

Derivative will be defined as $\frac{\Delta\>V}{\Delta\>r}$

Taking two general points on the graph

$(r, 2nr^3)$ and $(r+\Delta\>r,2\pi[r+\Delta\>r]^3)$

$\frac{\Delta\>V}{\Delta\>r}=\frac{2\pi[r+\Delta\>r]^3-2\pi\>r^3}{(r+\Delta\>r)-r}$

$=\frac{2\pi[r^3+3r^2\Delta\>r+3r(\Delta\>r)^2+(\Delta\>r)^3-r^3]}{\Delta\>r}$

$=2\pi[3r^2+3r\Delta\>r+\Delta\>r^2]$

As $\Delta\>r\to0,$

Then $\frac{\Delta\>V}{\Delta \>r}=6\pi\>r^2$