Just when I thought I understood taking derivatives, a textbook example of taking a derivative the equation $$[x(t)]^2+4000^2=[s(t)]^2$$ with respect to time shows that the derivative is $$x\frac{dx}{dt} = s\frac{ds}{dt}$$
However, after applying the power rule and the chain rule, I come up with $$2x(t)\cdot x'(t)=2s(t)\cdot s'(t)$$.
What concept am I missing about taking the derivative with respect to time? To elaborate a bit, I'm confused as to where the $\mathbf{x}$ and the $\mathbf{s}$ come from in $$\mathbf{x}\frac{dx}{dt} = \mathbf{s}\frac{ds}{dt}$$ from the example. Furthermore, to me it seems like they forgot to apply the power rule.
Example: An Airplane Flying at a Constant Elevation https://cnx.org/contents/[email protected]:74vQD30u@6/Related-Rates
Best Answer
You are not missing anything.
$$2x(t)\cdot x'(t)=2s(t)\cdot s'(t)$$
$$ \iff x(t)\cdot x'(t)=s(t)\cdot s'(t)$$
$$\iff x\frac{dx}{dt} = s\frac{ds}{dt}$$
They divided by $2$ and used the $ \frac{dx}{dt}$ for $x'(t)$.