`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 `

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`