Given q'(r) = -r^(-1/2), what is a possible expression for q(r)?

Answer 1

We've got our independent variable raised to a power here, so we can just think about the power rule!

If in general I know the derivative of x^n is n*x^(n-1), and here we're in the situation where our n-1=-1/2, we can pretty quickly see that the original power must have been 1/2

Oh, but there isn't a factor of 1/2 in front of the expression of q', just a -1. No problem, that just means that whatever was there multiplied by the 1/2 to become -1; that leads us to understand the coefficient in front of our r^(1/2) must have been -2

Sanity checking this answer: if q(r)=-2*r^(1/2), the derivative is -2*(1/2)*r^(-1/2), which indeed simplifies to -r^(-1/2)

Hopefully this helps you not just understand this specific problem, but how to more easily approach anti-derivatives of power functions in general!