A calculus of the absurd

11.10 Derivatives of trigonometric functions

11.10.1 Derivative of \(\sin (x)\)

What is the derivative of \(\sin (x)\)? First, we can use the definition of the limit and a little algebra.

\begin{align} \frac {d}{dx}[\sin (x)] &= \lim _{h\to 0} \frac {\sin (x+h) - \sin (x)}{h} \\ &= \lim _{h\to 0} \frac {\sin (x)\cos (h) + \sin (h)\cos (x) - \sin (x)}{h} \end{align}

We want to rewrite this in terms of the two limits we found in the previous section!

\begin{align*} \lim _{h\to 0} \frac {\sin (x)\cos (h) + \sin (h)\cos (x) - \sin (x)}{h} &= \lim _{h\to 0} \left [\frac {\sin (x)(\cos (h)-1)}{h} + \frac {\sin (h)}{h}\cos (x)\right ] \\ &= \lim _{h\to 0} \left [-\sin (x)\frac {(1-\cos (h))}{h} + \frac {\sin (h)}{h}\cos (x)\right ] \\ &= \lim _{h\to 0} \left [0 + 1\cos (x)\right ] \\ &= \cos (x) \end{align*}