A calculus of the absurd

10.10 Derivatives of trigonometric functions

10.10.1 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*}