How do I us the Limit definition of derivative on #f(x)=cos(x)#?

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Answer 1 · 2018-07-28T23:48:19

Use the general formula for the limit definition of the derivative.
You'll need to know the trigonometric addition formula and some limits.

We know that the formula for the limit definition of the derivative is:

#lim_{Deltax to 0}{f(x+Deltax)-f(x)}/{Deltax}#

So let's apply it:

#lim_{Deltax to 0}{cos(x+Deltax)-cosx}/{Deltax}#

#=lim_{Deltax to 0}{cosx*cosDeltax-sinx*sinDeltax-cosx}/{Deltax}#

#=lim_{Deltax to 0}{cosx(cosDeltax-1)-sinx*sinDeltax}/{Deltax}#

#=lim_{Deltax to 0}(cosx*{cosDeltax-1}/{Deltax}-sinx*{sinDeltax}/{Deltax})#

#=(1)(0)-(sinx)(1)#

#=-sinx#