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

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Answer 1 · 2016-12-28T09:54:04

#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#