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 | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{h \mathop \to 0} \frac {\map \cos {\paren {x + \frac h 2} + \frac h 2} - \map \cos {\paren {x + \frac h 2} - \frac h 2} } h\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{h \mathop \to 0} \frac {-2 \map \sin {x + \frac h 2} \map \sin {\frac h 2} } h\) | Werner 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 | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\sin x\) |
$\blacksquare$
Also see
Sources
- 1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $5$.
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation
- 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $5$. Differential Calculus: Appendix: Derivatives of fundamental functions: $3.$ Trigonometric functions
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): cosine
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Appendix $2$: Table of derivatives and integrals of common functions: Trigonometric functions
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $1$: Derivatives
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $1$: Derivatives
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $6$: Derivatives
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $7$: Derivatives