Answer to Question #264280 in Calculus for ella

Question #264280

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


1
Expert's answer
2021-11-17T15:42:20-0500

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"


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