We know that we can use the limit definition
to compute the derivative of “from first principles”. And this limit can be evaluated analytically in several ways. We could for example use the addition law for sine and write the limit as
and then compute the individual limits. This is where that fine old mathematical chestnut
Or we could use the sine difference formula
and so write the limit as
Neither of these approaches is wrong, but as teaching tools they leave a lot to be desired. They are very abstract, and several removes from the geometric notion of the derivative as a rate of change, or of the gradient of a tangent. Beginning students, especially those with weak backgrounds, are likely to be bemused.
So here is a geometric approach. I discovered that it is in fact very well known (well, something so simple was bound to be), but in fact I’d never come across this until today. So here it is.
Start with and on a diagram:
Then is the vertical side of the small blue triangle in this next diagram:
Since is very small, the hypotenuse of this triangle will be very close to the arc length of the circle, which is also , since we are working in radians. Thus the hypotenuse can be considered to be .
Now let’s look at a few angles. First the outer angles in the triangle created by will be equal, since the triangle is isosceles (the two long sides are both radii of the circle):
Both these angles will be equal to
Since the upper green angle in the first diagram is
the upper angle in the blue triangle will be
which simplifies to
Just to make this quite clear, here’s the top point:
as we found above.
So, the blue triangle has hypotenuse , vertical side and upper angle :
This means that, for small :
or alternatively that
Taking the limit of both sides as tends to zero:
Note that the base of the blue triangle, checking with the first diagram, is
This means that
from which, by taking limits of both sides as tends to zero, we obtain