Change in volume of sphere given change in radius

Question

Finding the change in volume $$V=\frac{4}{3}\pi a^3$$ of a sphere when the radius change from $a_{0}$ to $a_{0}+da$

What I tried:

Using differential formula

$$\frac{\Delta V}{\Delta a}=\frac{d V}{da}=\frac{d}{da}\bigg(\frac{4}{3}\pi a^3\bigg)=4\pi a^2$$

$$\Delta V=4\pi a^2 da$$

Is my answer is right.actually i dont have solution. If not Then how do i solve it. Thanks

Answer 1

$$\begin{align}\Delta V&=\frac 43\pi(a_0+da)^3-\frac 43\pi a_0^3\\&=\frac 43\pi((a_0+da)^3-a_0^3)\\&=\frac 43 \pi((a_0+da)-a_0)((a_0+da)^2+a_0(a_0+da)+a_0^2)\\&= \frac43\pi da(3a_0^2+3a_0da+da^2)\end{align}$$ If $|da|\ll a_0$ then $\Delta V=4\pi a_0^2 da$.

Answer 2

For this kinf of problems (they are very frequent in physics), you can make it much faster using logarithmic differentiation $$V=\frac{4}{3}\pi a^3 \implies \log(V)=\log\left(\frac{4}{3}\pi \right)+3\log(a)$$

$$\frac{dV}{V}=3\frac {da}a\implies\frac{\Delta V}{V}=3\frac {\Delta a}a\implies \Delta V=\frac{4}{3}\pi a^3\times3\frac {\Delta a}a=4\pi a^2\Delta a$$