To solve for $r'$, we need to differentiate $n = \frac{1 + r^2}{2}$ with respect to $t$ (assuming $r$ and $n$ are functions of $t$).

1. Start with the equation: $n = \frac{1 + r^2}{2}$

2. Differentiate both sides with respect to $t$: $\frac{dn}{dt} = \frac{d}{dt} \left( \frac{1 + r^2}{2} \right)$

3. Applying the chain rule on the right side, we get: $\frac{dn}{dt} = \frac{1}{2} \cdot 2r \cdot \frac{dr}{dt} = r \cdot \frac{dr}{dt}$

4. Let $\frac{dn}{dt} = n'$ and $\frac{dr}{dt} = r'$. Substitute these into the equation: $n' = r \cdot r'$

5. Solve for $r'$: $r' = \frac{n'}{r}$

Thus, the derivative $r'$ is: $r' = \frac{n'}{r}$

Let me know if you’d like more details on any step.

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Here are some related questions for further exploration:

1. What if $r$ is not a function of $t$, but of a different variable?
2. How would you solve for $r''$, the second derivative of $r$?
3. Can we interpret $r'$ geometrically in this context?
4. How would the solution change if we differentiated with respect to another variable?
5. What applications does this type of differentiation have in physics or engineering?

Tip: When differentiating, always identify which variables are functions of time (or another variable) to avoid errors.