I have a known derivative given as:
$\frac{dr}{dt} = \frac{a}{r^2}$
And then from that information, I am trying to find:
$\frac{d(r^2)}{dt}$
I know that this is equal to $\frac{dr^2}{dr} \times \frac{dr}{dt}$
Which gives by the chain rule:
$\frac{2a}{r}$
But its really not obvious to me why the chain rule is done here in order to solve it. Hope some one can explain it better than what I have so far read online, it is very confusing.
I take the chain rule for a function within a function, but i am not seeing how this is the case for $r^2$.
Best Answer
Let $f$ be a differentiable function and put $g(x):=f(x)^2.$ Then $g(x)=h(f(x)),$ with $h(x)=x^2.$
The chain rule gives:
$$g'(x)=h'(f(x)) f'(x)=2f(x)f'(x).$$