# Derivative of Cosine Function

## Theorem

$D_x \left({\cos x}\right) = -\sin x$

### Corollary

$D_x \left({\cos a x}\right) = -a \sin a x$

## Proof 1

From the definition of the cosine function, we have:

$\displaystyle \cos x = \sum_{n \mathop = 0}^\infty \left({-1}\right)^n \frac {x^{2n}}{\left({2n}\right)!}$

Then:

 $\displaystyle D_x \left({\cos x}\right)$ $=$ $\displaystyle \sum_{n \mathop = 1}^\infty \left({-1}\right)^n 2n \frac {x^{2n - 1} }{\left({2n}\right)!}$ Power Series is Differentiable on Interval of Convergence $\displaystyle$ $=$ $\displaystyle \sum_{n \mathop = 1}^\infty \left({-1}\right)^n \frac {x^{2n - 1} }{\left({2n - 1}\right)!}$ $\displaystyle$ $=$ $\displaystyle \sum_{n \mathop = 0}^\infty \left({-1}\right)^{n+1} \frac {x^{2n + 1} }{\left({2n + 1}\right)!}$ changing summation index $\displaystyle$ $=$ $\displaystyle -\sum_{n \mathop = 0}^\infty \left({-1}\right)^n \frac {x^{2n + 1} }{\left({2n + 1}\right)!}$

The result follows from the definition of the sine function.

$\blacksquare$

## Proof 2

 $\displaystyle D_x \left({\cos x}\right)$ $=$ $\displaystyle \lim_{h \mathop \to 0} \frac {\cos \left({x + h}\right) - \cos \left({x}\right)} h$ Definition of Derivative of Real Function at Point $\displaystyle$ $=$ $\displaystyle \lim_{h \mathop \to 0} \frac {\cos \left({x}\right) \cos \left({h}\right) - \sin \left({x}\right) \sin \left({h}\right) - \cos \left({x}\right)} h$ Cosine of Sum $\displaystyle$ $=$ $\displaystyle \lim_{h \mathop \to 0} \frac {\cos \left({x}\right) \cos \left({h}\right) - \cos \left({x}\right)} h + \lim_{h \mathop \to 0} \frac{- \sin \left({x}\right) \sin \left({h}\right)} h$ Sum Rule for Limits of Functions $\displaystyle$ $=$ $\displaystyle \cos \left({x}\right) \ \lim_{h \mathop \to 0} \frac {\cos \left({h}\right) - 1} h - \sin \left({x}\right) \ \lim_{h \mathop \to 0} \frac {\sin \left({h}\right)} h$ Multiple Rule for Limits of Functions $\displaystyle$ $=$ $\displaystyle \cos \left({x}\right) \times 0 - \sin \left({x}\right) \times 1$ Limit of (Cosine (X) - 1) over X and Limit of Sine of X over X $\displaystyle$ $=$ $\displaystyle - \sin \left({x}\right)$

$\blacksquare$

## Proof 3

 $\displaystyle D_x \cos x$ $=$ $\displaystyle D_x \, \map \sin {\frac \pi 2 - x}$ Sine of Complement equals Cosine $\displaystyle$ $=$ $\displaystyle -\map \cos {\frac \pi 2 - x}$ Derivative of Sine Function and Chain Rule $\displaystyle$ $=$ $\displaystyle -\sin x$ Cosine of Complement equals Sine

$\blacksquare$

## Proof 4

 $\displaystyle D_x \left({\cos x}\right)$ $=$ $\displaystyle \lim_{h \mathop \to 0} \frac {\cos \left({x + h}\right) - \cos \left({x}\right)} h$ Definition of Derivative of Real Function at Point $\displaystyle$ $=$ $\displaystyle \lim_{h \mathop \to 0} \frac {\cos \left({\left({x + \frac h 2}\right) + \frac h 2}\right) - \cos \left({\left({x + \frac h 2}\right) - \frac h 2}\right)} h$ $\displaystyle$ $=$ $\displaystyle \lim_{h \mathop \to 0} \frac {-2 \sin \left({x + \frac h 2}\right) \sin \left({\frac h 2}\right)} h$ Simpson's Formula for Sine by Sine $\displaystyle$ $=$ $\displaystyle -\lim_{h \mathop \to 0} \sin \left({x + \frac h 2}\right) \lim_{h \mathop \to 0} \frac{\sin \left({\frac h 2}\right)} {\frac h 2}$ Multiple Rule for Limits of Functions and Product Rule for Limits of Functions $\displaystyle$ $=$ $\displaystyle -\sin \left({x}\right) \times 1$ Continuity of Sine and Limit of Sine of X over X $\displaystyle$ $=$ $\displaystyle -\sin \left({x}\right)$

$\blacksquare$