d d x ( sin ( x ) ) = cos ( x ) {\displaystyle \frac{d}{dx}(\sin(x))=\cos(x)}
Let f ( x ) = sin ( x ) {\displaystyle f(x) = \sin(x)} . Then:
Using addition trigonometric identities,
Since
. Then:
![{\displaystyle {\begin{aligned}f'(x)&=\lim _{h\to 0}{\frac {\sin(x)\cos(h)+\cos(x)\sin(h)-\sin(x)}{h}}\\&=\lim _{h\to 0}\left[\cos(x){\frac {\sin(h)}{h}}+\sin(x){\frac {\cos(h)-1}{h}}\right]\\&=\lim _{h\to 0}\left[\cos(x){\frac {\sin(h)}{h}}\right]+\lim _{h\to 0}\left[\sin(x){\frac {\cos(h)-1}{h}}\right]\\&=\lim _{h\to 0}\cos(x)\cdot \lim _{h\to 0}{\frac {\sin(h)}{h}}+\lim _{h\to 0}\sin(x)\cdot \lim _{h\to 0}{\frac {\cos(h)-1}{h}}\\&=\cos(x)\cdot \lim _{h\to 0}{\frac {\sin(h)}{h}}+\sin(x)\cdot \lim _{h\to 0}{\frac {\cos(h)-1}{h}}\end{aligned}}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/af7f3f4bbb1347c44dc2d4a607b66065398d8a03)
