Derivative of Cosine Function

Theorem

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

Corollary

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

Proof 1

From the definition of the cosine function, we have $\displaystyle \cos x = \sum_{n=0}^\infty \left({-1}\right)^n \frac {x^{2n}}{\left({2n}\right)!}$.

Then:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle D_x \left({\cos x}\right)\)	\(=\)	\(\displaystyle \sum_{n=1}^\infty \left({-1}\right)^n 2n \frac {x^{2n - 1} }{\left({2n}\right)!}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Power Series Differentiable on Interval of Convergence
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sum_{n=1}^\infty \left({-1}\right)^n \frac {x^{2n - 1} }{\left({2n - 1}\right)!}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sum_{n=0}^\infty \left({-1}\right)^{n+1} \frac {x^{2n + 1} }{\left({2n + 1}\right)!}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Changing summation index
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle - \sum_{n=0}^\infty \left({-1}\right)^n \frac {x^{2n + 1} }{\left({2n + 1}\right)!}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

The result follows from the definition of the sine function.

$\blacksquare$

Proof 2

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

$\blacksquare$

Proof 3

This proof depends on Derivative of Sine Function.

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle D_x \cos x\)	\(=\)	\(\displaystyle D_x \sin \left({\frac \pi 2 - x}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Sine and Cosine of Complementary Angles
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle -\cos \left({\frac \pi 2 - x}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Derivative of Sine Function and Chain Rule
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle - \sin x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	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): $13.15$
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 16.3 \ (1) \ \text{(v)}$