# Question 431bc

Feb 12, 2018

$- \sin x$

#### Explanation:

${\lim}_{\delta \to 0} \frac{\cos \left(x + \delta\right) - \cos \left(x\right)}{\delta} = - \sin x$

I can't prove it.

Feb 12, 2018

See explanation

#### Explanation:

There are multiple ways to show this

A common way is to use the definition of a derivative,
which is nicely done here on the site

Another way is to use series expansions

Cosine and sine can expressed as the series expansions

• cos(x)=1/(0!)-x^2/(2!)+x^4/(4!)-x^6/(6!)...

• sin(x)=x/(1!)-x^3/(3!)+x^5/(5!)-x^7/(7!)...

Take the derivative of cosine

d/dxcos(x)=d/dx(1/(0!)-x^2/(2!)+x^4/(4!)-x^6/(6!)...)

=-2x/(2!)+4x^3/(4!)-6x^5/(6!)...

=-x/(1!)+x^3/(3!)-x^5/(5!)...

=-(x/(1!)-x^3/(3!)+x^5/(5!)...)#

$= - \sin \left(x\right)$