How do you use the definition of a derivative to find the derivative of $f (x) = cos x$ $f(x)=cosx$ ?

Question

1480 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-12-28T09:54:04

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

By definition:

$f'(x) = lim_(Deltax->0) (f(x+Deltax)-f(x))/(Deltax)$

In our case, $f(x) = cosx$ , so:

$d/(dx) cosx= lim_(Deltax->0) (cos(x+Deltax)-cos(x))/(Deltax)$

We can now use the trigonometric identity:

$cos(x+Deltax) = cosxcos(Deltax)-sinxsin(Deltax)$

and obtain:

$d/(dx) cosx=lim_(Deltax->0)(cosxcos(Deltax)-sinxsin(Deltax)-cosx)/(Deltax)$

that is:

$d/(dx) cosx=lim_(Deltax->0)cosx(1-cos(Deltax))/(Deltax)-sinx(sin(Deltax))/(Deltax)$

But:

$lim_(t->0) sint/t=1$

and:

$lim_(t->0) (1-cost)/t=0$

and we can conclude:

$d/(dx) cosx=-sinx$

How do you use the definition of a derivative to find the derivative of f(x)=cosxf(x)=cosxf(x)=cosx?