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:

(1)

Using (1) to expand  gives

(2)

(3)

Using (1) to expand  to an imaginary power, where  gives

(4)

Rearranging (4) we have:

Comparing this with (2) and (3) gives:

Here is another example:

Show that

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,

.                                                                     (1)

Then the magnitude of the derivative is

and rearranging gives

.                                                 (2)

Using (1) in (2) gives