Attempting to find the maximum curvature of the curve $e^x$ I have to differentiate kappa $$k(t)=\frac{e^t}{(1+e^t)^{3/2}}$$
of course, I use quotient rule and chain rule to find the derivative;
$$k^1(t)=\frac{e^t*(1-e^{2t})^{3/2}-e^t*3/2(1-e^{2t})*2e^{2t}}{((1+e^{2t})^{3/2})^2}$$
and simplify etc.
However, multiple sources including an exam key, and even a highly upvoted post on Stack Exchange Math has the derivative of kappa as
$$k^1(t)=\frac{e^t*(1-e^{2t})^{3/2}-e^t*3/2(1-e^{2t})*2t}{((1+e^{2t})^{3/2})^2}$$
Notice the blatant abuse of chain rule to get the last factor in the numerator.
Why are they doing this? am I missing something? are these people blatantly wrong? please help
If you don't believe me look at these pages:
question # 4 http://www.math.washington.edu/~conroy/m126-general/exams/mt2SolMath126Win2006.pdf
And look at chris' post: How to find the point where the curvature is maximum
fixed the typos, thanks to Lee and Mattos for pointing them out~