Derivative of Cosine Function

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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
\(\displaystyle \) \(=\) \(\displaystyle -\sin \left({x}\right)\)

$\blacksquare$


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