Derivative of Sine Function

Theorem

$D_x \left({\sin x}\right) = \cos x$

Corollary

$D_x \left({\sin \left({a x}\right)}\right) = a \cos \left({a x}\right)$

Proof 1

From the definition of the sine function, we have:

$\displaystyle \sin x = \sum_{n=0}^\infty \left({-1}\right)^n \frac {x^{2n+1}} {\left({2n+1}\right)!}$

From Power Series over Factorial, this series converges for all $x$.

From Power Series Differentiable on Interval of Convergence, we have:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle D_x \left({\sin x}\right)\)	\(=\)	\(\displaystyle \sum_{n=0}^\infty \left({-1}\right)^n \left({2n+1}\right) \frac {x^{2n} } {\left({2n+1}\right)!}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sum_{n=0}^\infty \left({-1}\right)^n \frac {x^{2n} } {\left({2n}\right)!}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

The result follows from the definition of the cosine function.

$\blacksquare$

Proof 2

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle D_x \left({\sin x}\right)\)	\(=\)	\(\displaystyle \lim_{h \to 0} \frac { \sin \left({x + h}\right) - \sin \left({x}\right) } h\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of derivative
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \lim_{h \to 0} \frac { \sin \left({x}\right) \cos \left({h}\right) + \sin \left({h}\right) \cos \left({x}\right) - \sin \left({x}\right) } h\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Angle Addition and Subtraction Formulas
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \lim_{h \to 0} \frac { \sin \left({x}\right) \left({\cos \left({h}\right) - 1}\right) + \sin \left({h}\right) \cos \left({x}\right) } h\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Collecting terms containing $\sin \left({x}\right)$ and factoring
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \lim_{h \to 0} \frac { \sin \left({x}\right) \left({\cos \left({h}\right) - 1}\right) } h + \lim_{h \to 0} \frac { \sin \left({h}\right) \cos \left({x}\right) } h\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Sum Rule
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sin \left({x}\right) \times 0 + 1 \times \cos \left({x}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	From Limit of Sine of X over X and Limit of (Cosine (X) - 1) over X
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \cos \left({x}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

$\blacksquare$

Proof 3

This proof depends on Derivative of Cosine Function.

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle D_x \sin x\)	\(=\)	\(\displaystyle D_x \cos \left({\frac \pi 2 - x}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Sine and Cosine of Complementary Angles
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sin \left({\frac \pi 2 - x}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Derivative of Cosine Function and Chain Rule
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \cos x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Sine and Cosine of Complementary Angles

$\blacksquare$

Proof of Corollary

Follows directly from Derivative of Function of Constant Multiple.

$\blacksquare$

Sources

• Murray R. Spiegel: Mathematical Handbook of Formulas and Tables (1968): $13.14$
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 16.3 \ (1) \ \text{(vi)}$