Derivative of Sine Function
Theorem
- $\map {\dfrac \d {\d x} } {\sin x} = \cos x$
Corollary
- $\map {\dfrac \d {\d x} } {\sin a x} = a \cos a x$
Proof 1
From the definition of the sine function, we have:
- $\ds \sin x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}$
From Radius of Convergence of Power Series over Factorial, this series converges for all $x$.
From Power Series is Differentiable on Interval of Convergence:
| \(\ds \map {\frac \d {\d x} } {\sin x}\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!}\) |
The result follows from the definition of the cosine function.
$\blacksquare$
Proof 2
| \(\ds \map {\frac \d {\d x} } {\sin x}\) | \(=\) | \(\ds \lim_{h \mathop \to 0} \frac {\map \sin {x + h} - \sin x} h\) | Definition of Derivative of Real Function at Point | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{h \mathop \to 0} \frac {\sin x \cos h + \sin h \cos x - \sin x} h\) | Sine of Sum | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{h \mathop \to 0} \frac {\sin x \paren {\cos h - 1} + \sin h \cos x} h\) | collecting terms containing $\map \sin x$ and factoring | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{h \mathop \to 0} \frac {\sin x \paren {\cos h - 1} } h + \lim_{h \mathop \to 0} \frac {\sin h \cos x} h\) | Sum Rule for Limits of Real Functions | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \sin x \times 0 + 1 \times \cos x\) | Limit of $\dfrac {\sin x} x$ at Zero and Limit of $\dfrac {\cos x - 1} x$ at Zero | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos x\) |
$\blacksquare$
Proof 3
| \(\ds \dfrac \d {\d x} \sin x\) | \(=\) | \(\ds \dfrac \d {\d x} \map \cos {\frac \pi 2 - x}\) | Cosine of Complement equals Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \sin {\frac \pi 2 - x}\) | Derivative of Cosine Function and Chain Rule for Derivatives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos x\) | Sine of Complement equals Cosine |
$\blacksquare$
Proof 4
| \(\ds \map {\frac \d {\d x} } {\sin x}\) | \(=\) | \(\ds \lim_{h \mathop \to 0} \frac {\map \sin {x + h} - \sin x} h\) | Definition of Derivative of Real Function at Point | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{h \mathop \to 0} \frac {\map \sin {\paren {x + \frac h 2} + \frac h 2} - \map \sin {\paren {x + \frac h 2} - \frac h 2} } h\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{h \mathop \to 0} \frac {2 \map \cos {x + \frac h 2} \map \sin {\frac h 2} } h\) | Werner Formula for Cosine by Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{h \mathop \to 0} \map \cos {x + \frac h 2} \lim_{h \mathop \to 0} \frac {\map \sin {\frac h 2} } {\frac h 2}\) | Multiple Rule for Limits of Real Functions and Product Rule for Limits of Real Functions | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos x \times 1\) | Cosine Function is Continuous and Limit of $\dfrac {\sin x} x$ at Zero | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos x\) |
$\blacksquare$
Proof 5
| \(\ds \map \arcsin x\) | \(=\) | \(\ds \int_0^x \frac {\d x} {\sqrt {1 - x^2} }\) | Arcsine as Integral | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \dfrac {\map \d {\map \arcsin y} } {\d y}\) | \(=\) | \(\ds \dfrac {\map \d {\ds \int_0^y \dfrac 1 {\sqrt {1 - y^2} } \rd y} } {\d y}\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {\sqrt {1 - y^2} }\) | Corollary to Fundamental Theorem of Calculus: First Part |
Note that we get the same answer as Derivative of Arcsine Function.
By definition of real $\arcsin$ function, $\arcsin$ is bijective on its domain $\closedint {-1} 1$.
Thus its inverse is itself a mapping.
From Inverse of Inverse of Bijection, its inverse is the $\sin$ function.
| \(\ds \dfrac {\map \d {\sin \theta} } {\d \theta}\) | \(=\) | \(\ds 1 / \dfrac 1 {\sqrt {1 - \sin^2 \theta} }\) | Derivative of Inverse Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \pm \sqrt {1 - \sin^2 \theta}\) | Positive in Quadrant $\text I$ and Quadrant $\text {IV}$, Negative in Quadrant $\text {II}$ and Quadrant $\text {III}$ | |||||||||||
| \(\ds \dfrac {\map \d {\sin \theta} } {\d \theta}\) | \(=\) | \(\ds \cos \theta\) |
$\blacksquare$
Also see
Sources
- 1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $4$.
- 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): sine
- 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