What is the derivative of:
2pi^2r^4
would it simply be 8pi^2r^3?
I was thinking the 2pi^2 is simply the equivalent of a constant in front of the r^4 and so you would just use the power rule.
To find the derivative of 2π^2r^4, you can use the power rule for derivatives. The power rule states that for a term of the form x^n, the derivative is nx^(n-1).
Applying the power rule, the derivative of r^4 is 4r^(4-1) = 4r^3.
Since 2π^2 is a constant, the derivative of 2π^2r^4 is simply the product of the derivative of the variable term (4r^3) and the constant term (2π^2). So the derivative of 2π^2r^4 is:
2π^2 * 4r^3 = 8π^2r^3.
Therefore, your initial intuition is correct, and the derivative of 2π^2r^4 is indeed 8π^2r^3.