derivatives of $sinx$ and $cosx$

Theorem 1.

$$\frac{d}{dx}(\sin x)=\cos x$$

Proof.

${\displaystyle \frac{d}{dx}}(\sin x)$	$=\underset{h\to 0}{lim}{\displaystyle \frac{\sin (x+h)-\sin x}{h}}$
	$=\underset{h\to 0}{lim}{\displaystyle \frac{\sin x\cos h+\cos x\sin h-\sin x}{h}}$ by addition formula for sine
	$=\underset{h\to 0}{lim}{\displaystyle \frac{\sin x(\cos h-1)+\cos x\sin h}{h}}$
	$=\underset{h\to 0}{lim}\left(\sin x\cdot {\displaystyle \frac{\cos h-1}{h}}+\cos x\cdot {\displaystyle \frac{\sin h}{h}}\right)$
	$=\sin x\left(\underset{h\to 0}{lim}{\displaystyle \frac{\cos h-1}{h}}\right)+\cos x\left(\underset{h\to 0}{lim}{\displaystyle \frac{\sin h}{h}}\right)$ by this entry (http://planetmath.org/LimitRulesOfFunctions)
	$=\sin x\cdot 0+\cos x\cdot 1$ by this theorem (http://planetmath.org/LimitOfDisplaystyleFracsinXxAsXApproaches0) and its corollary (http://planetmath.org/LimitOfDisplaystyleFrac1CosXxAsXApproaches0)
	$=\cos x$

∎

Theorem 2.

$$\frac{d}{dx}(\cos x)=-\sin x$$

Proof.

${\displaystyle \frac{d}{dx}}(\cos x)$	$=\underset{h\to 0}{lim}{\displaystyle \frac{\cos (x+h)-\cos x}{h}}$
	$=\underset{h\to 0}{lim}{\displaystyle \frac{\cos x\cos h-\sin x\sin h-\cos x}{h}}$ by addition formula for cosine (http://planetmath.org/AdditionFormulaForCosine)
	$=\underset{h\to 0}{lim}{\displaystyle \frac{\cos x(\cos h-1)+\sin x\sin h}{h}}$
	$=\underset{h\to 0}{lim}\left(\cos x\cdot {\displaystyle \frac{\cos h-1}{h}}-\sin x\cdot {\displaystyle \frac{\sin h}{h}}\right)$
	$=\cos x\left(\underset{h\to 0}{lim}{\displaystyle \frac{\cos h-1}{h}}\right)-\sin x\left(\underset{h\to 0}{lim}{\displaystyle \frac{\sin h}{h}}\right)$ by this entry (http://planetmath.org/LimitRulesOfFunctions)
	$=\cos x\cdot 0-\sin x\cdot 1$ by this theorem (http://planetmath.org/LimitOfDisplaystyleFracsinXxAsXApproaches0) and its corollary (http://planetmath.org/LimitOfDisplaystyleFrac1CosXxAsXApproaches0)
	$=-\sin x$

∎

Title	derivatives of $sinx$ and $cosx$
Canonical name	DerivativesOfsinXAndcosX
Date of creation	2013-03-22 16:58:51
Last modified on	2013-03-22 16:58:51
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	8
Author	Wkbj79 (1863)
Entry type	Theorem
Classification	msc 26A06
Classification	msc 26A09
Classification	msc 26A03
Related topic	Derivative2
Related topic	LimitOfDisplaystyleFracsinXxAsXApproaches0
Related topic	LimitOfDisplaystyleFrac1CosXxAsXApproaches0
Related topic	DerivativesOfSineAndCosine