Derivative of Cosine Function

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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:

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

Then:

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


The result follows from the definition of the sine function.

$\blacksquare$


Proof 2

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

$\blacksquare$


Proof 3

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

$\blacksquare$


Proof 4

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

$\blacksquare$


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