Part 3/6:
If we were to sketch the derivative of the sine function based on the previously identified slopes, we would produce a curve that reflects the changes in the sine function.
For instance, since the derivative is zero at maximum and minimum points of sine, we would see corresponding points at these locations on the derivative graph. When ( \sin x ) is increasing, we observe positive values for the derivative, leading us to conclude that the derivative function resembles the cosine function, ( \cos x ). This sets the stage for our formal proof.
The Definition of the Derivative
Next, let’s pursue a formal derivation of the derivative of sine using its definition. The derivative of a function ( f ) at a point ( x ) is defined as:
[
f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}