# Derivative of Cosine Function

## Theorem

$\map {\dfrac \d {\d x} } {\cos x} = -\sin x$

### Corollary

$\map {\dfrac \d {\d x} } {\cos a x} = -a \sin a x$

## Proof 1

From the definition of the cosine function, we have:

$\ds \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!}$

Then:

 $\ds \map {\frac \d {\d x} } {\cos x}$ $=$ $\ds \sum_{n \mathop = 1}^\infty \paren {-1}^n 2 n \frac {x^{2 n - 1} } {\paren {2 n}!}$ Power Series is Differentiable on Interval of Convergence $\ds$ $=$ $\ds \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {x^{2 n - 1} } {\paren {2 n - 1}!}$ $\ds$ $=$ $\ds \sum_{n \mathop = 0}^\infty \paren {-1}^{n + 1} \frac {x^{2 n + 1} } {\paren {2 n + 1}!}$ changing summation index $\ds$ $=$ $\ds -\sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}$

The result follows from the definition of the sine function.

$\blacksquare$

## Proof 2

 $\ds \map {\frac \d {\d x} } {\cos x}$ $=$ $\ds \lim_{h \mathop \to 0} \frac {\map \cos {x + h} - \cos x} h$ Definition of Derivative of Real Function at Point $\ds$ $=$ $\ds \lim_{h \mathop \to 0} \frac {\cos x \cos h - \sin x \sin h - \cos x} h$ Cosine of Sum $\ds$ $=$ $\ds \lim_{h \mathop \to 0} \frac {\cos x \cos h - \cos x} h + \lim_{h \mathop \to 0} \frac {-\sin x \sin h} h$ Sum Rule for Limits of Real Functions $\ds$ $=$ $\ds \cos x \lim_{h \mathop \to 0} \frac {\cos h - 1} h - \sin x \lim_{h \mathop \to 0} \frac {\sin h} h$ Multiple Rule for Limits of Real Functions $\ds$ $=$ $\ds \cos x \times 0 - \sin x \times 1$ Limit of $\dfrac {\cos x - 1} x$ at Zero and Limit of $\dfrac {\sin x} x$ at Zero $\ds$ $=$ $\ds -\sin x$

$\blacksquare$

## Proof 3

 $\ds \frac \d {\d x} \cos x$ $=$ $\ds \frac \d {\d x} \map \sin {\frac \pi 2 - x}$ Sine of Complement equals Cosine $\ds$ $=$ $\ds -\map \cos {\frac \pi 2 - x}$ Derivative of Sine Function and Chain Rule for Derivatives $\ds$ $=$ $\ds -\sin x$ Cosine of Complement equals Sine

$\blacksquare$

## Proof 4

 $\ds \map {\frac \d {\d x} } {\cos x}$ $=$ $\ds \lim_{h \mathop \to 0} \frac {\map \cos {x + h} - \cos x} h$ Definition of Derivative of Real Function at Point $\ds$ $=$ $\ds \lim_{h \mathop \to 0} \frac {\map \cos {\paren {x + \frac h 2} + \frac h 2} - \map \cos {\paren {x + \frac h 2} - \frac h 2} } h$ $\ds$ $=$ $\ds \lim_{h \mathop \to 0} \frac {-2 \map \sin {x + \frac h 2} \map \sin {\frac h 2} } h$ Simpson's Formula for Sine by Sine $\ds$ $=$ $\ds -\lim_{h \mathop \to 0} \map \sin {x + \frac h 2} \lim_{h \mathop \to 0} \frac {\map \sin {\frac h 2} } {\frac h 2}$ Multiple Rule for Limits of Real Functions and Product Rule for Limits of Real Functions $\ds$ $=$ $\ds -\sin x \times 1$ Real Sine Function is Continuous and Limit of $\dfrac {\sin x} x$ at Zero $\ds$ $=$ $\ds -\sin x$

$\blacksquare$