Taking derivatives of exponential function

Question

Beware, this question might be silly and may contain mathematical fallacies.

$$ d/dt(e^{jwt}) = jwe^{jwt} $$ $$ d/dt(e^{j \pi t}) = j \pi e^{j \pi t} $$ $$ d/dt(e^{j 180 t}) = j 180 e^{j 180 t} $$

In the second equation $\pi$ is in radians and in the third one 180 is in degrees. These two equations' derivatives must give the same results since they're the same. So are the units of these angles conserved when we take derivatives? In other words are the 180 and $\pi$ at the right hand sides of equations just numbers or numbers with units? Because otherwise the results seem different.

Answer 1

In the second equation pi is in radians and in the third one 180 is in degrees.

No they are not, both are dimensionless numbers.

Edit: Let $R$ denote a radius measured in meters, then the area of the disk of radius $R$ measured in square meters is $A=\pi R^2$. What is the unit of $\pi$? Well, $\pi=A/R^2$, the unit of $A$ is the square meter, the unit of $R^2$ is the square meter, hence...

Answer 2

$\pi$ is not equal to 180. ${}{}{}{}{}{}{}{}$

Try your reasoning on this one: My desk is 8 feet long. It's also 96 inches long. If I take the derivative of $x^8$, I get $8x^7$. If I take the derivative of $x^{96}$, I get $96x^{95}$. How can this be?