Answer in Calculus for ella #264280
January 25th, 2023
Use the definition of the derivative to differentiate V=(4)/(2)pi r^(3).
Rewriting the expression given
“V=frac{4}{2}pi>r^3”
“V=2pi>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,2pi[r+Delta>r]^3)”
“frac{Delta>V}{Delta>r}=frac{2pi[r+Delta>r]^3-2pi>r^3}{(r+Delta>r)-r}”
“=frac{2pi[r^3+3r^2Delta>r+3r(Delta>r)^2+(Delta>r)^3-r^3]}{Delta>r}”
“=2pi[3r^2+3rDelta>r+Delta>r^2]”
As “Delta>rto0,”
Then “frac{Delta>V}{Delta >r}=6pi>r^2”
