So a classic thing to derive in calculus textbooks is something like a statement as follows
Is $\frac{d}{dx}\sin(u)$ the same as the derivative of $\frac{d}{dx}\sin(x)$ where $u$ is an angle measured in degrees and $x$ is measured in radians? and of course the answer is no because of the chain rule.
Except usually this is ambiguously worded as "Is the derivative of $\sin(u)$, where $u$ is measured in degrees, equal to the derivative of $\sin(x)$ where $x$ is the same angle but measured in in radians?"
Then the texts go on to say something like "No and this why we don't work in degrees and instead chose to work in radians, to avoid all the messy constants that come out of taking derivatives." Am I crazy by thinking this is an odd thing to say that will end up confusing students. If your independent variable was an angle measured in degrees, you are probably more interested in it's derivative with respect to degrees not radians, which would infact be equal at the corresponding degrees and radians of an angle. Is my understanding wrong here. Is what the books say fine? I think at minimum they should at least be clear that we are taking the derivative with respect to radians, no?
Note this is not a duplicate of
Derivative of the sine function when the argument is measured in degrees
Even though it is highly related.
Best Answer
Whether an angle is measured in degrees or radians is most definitely worth bringing up in a class/lecture so the students can get a solid understanding of derivatives.
UNITS DO MATTER!!
Of course they do. Saying units do not matter is like claiming that $50$ miles per hour is equivalent to 50 feet per hour or 50 miles per second.
For x measured in radians, $d/dx(\sin x) = \cos x$
For x measured in degrees, $d/dx(\sin x) = (\pi/180) \cos x$
A derivative is the slope of a line, the change in the vertical units per change in the horizontal units. When you go from radians to degrees, the change in vertical units remains constant while the change in horizontal units increases tremendously (which, in turn, has the derivative become a lot smaller).
A good way to see this is simply to graph $\sin x$ in degrees. If the derivatives are all the same as they are when measured in radians you should find a tangent line with slope $1$ somewhere (as is true for $\sin x$ at $x=0$ radians). You will not find any tangents with slopes nearly this large when your angles are measured in degrees.
What you guys are claiming is "obvious" is patently false.