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`V'(r)` symbolizes the derivative of `V(r)` . When taking the derivative of a function, one must take note of the variable that is changing, which in this case is the variable "r". Therefore, the derivative of the function `V(r)=(4/3)*pi*r^3` will involve dropping the exponent that "r" is raised to, multiplying it with "r", and reducing...

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`V'(r)` symbolizes the derivative of `V(r)` . When taking the derivative of a function, one must take note of the variable that is changing, which in this case is the variable "r". Therefore, the derivative of the function `V(r)=(4/3)*pi*r^3` will involve dropping the exponent that "r" is raised to, multiplying it with "r", and reducing the exponent of "r" by 1. As a result:

`V(r) = (4/3)*pi*r^3` becomes `V'(r) = (4/3)*pi*3r^2`

(Notice that the exponent of 3 is now multiplying "r" and the exponent of "r" is reduced by 1. This is how one goes about taking a simple derivative. If there were more variables "r" in the function, the same method would be repeated on those "r"s too.)

Simplifying would cancel out the 3s, giving us the answer:

` V'(r)=4*pi*r^2 `

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The function `V(r) = (4/3)*pi*r^3` .

`V'(r) = (4/3)*pi*3*r^2 = 4*pi*r^2`

The derivative of `V(r) = (4/3)*pi*r^3` is `V'(r) = 4*pi*r^2`

Approved by eNotes Editorial Team