How to Do Physics Derivations
When learning a derivation, first write out (in a sentence) each step. It will help you follow the logic and it will help you to develop a style and framework for relations that you will later derive on your own.
Here is an example for the derivation of Euler's relation, :
1. Write out the general form of the Maclaurin series.
2. Use the Maclaurin series to expand: .
3. Use the Maclaurin series to expand
4. Rearrange the terms in the expansion of grouping the real terms together and the imaginary terms together.
5. Compare the expression of with the expansions of and and rewrite as a function of and .
The derivation will look like this:
Show that .
The general form of the Maclaurin series is:
Using (1) to expand gives
Using (1) to expand to an imaginary power, where gives
Rearranging (4) we have:
Comparing this with (2) and (3) gives:
Here is another example:
1. Write down the equation showing the relationship of wavelength, frequency and the speed of light.
2. Take the magnitude of the derivative with respect to n .
3. Separate and move one of the of over to the other side.
4. Substitute the relationship of for the on the side of the equation.
The actual derivation will look like this:
Show that .
For light waves,
Then the magnitude of the derivative is
and rearranging gives
Using (1) in (2) gives