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

1 Answer
Jul 28, 2018

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

Explanation:

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