`x`

is `cos(x)`

and for the cosine curve at x it is `-sin(x)`

.We start from

`x`

, and then move a little bit `h`

. Using the rule for sum of sines (here):The first term is

`-sin(x)`

times `(1/h)(1 - cos(h))`

; last time we showed that `(1/h)(1 - cos(h))`

equals zero in the limit as `h -> 0`

.The second term is

`cos(x)`

times `(1/h) sin(h)`

; we showed that `(1/h) sin(h)`

approaches `1`

as `h -> 0`

. Thus, Since sine and cosine are periodic with cosine "ahead"

If

Let

Then