Differentiating sin(x) from First Principles

Key Questions

  • How do you differentiate f(x)=sin(x)f(x)=sin(x) from first principles?

    Answer:

    d/dxsinx=cosxddxsinx=cosx

    Explanation:

    By definition of the derivative:

    f'(x)=lim_(h rarr 0) ( f(x+h)-f(x) ) / h

    So with f(x) = sinx we have;

    f'(x)=lim_(h rarr 0) ( sin(x+h) - sin x ) / h

    Using sin (A+B)=sinAcosB+sinBcosA we get

    f'(x)=lim_(h rarr 0) ( sinxcos h+sin hcosx - sin x ) / h

    \ \ \ \ \ \ \ \ \=lim_(h rarr 0) ( sinx(cos h-1)+sin hcosx ) / h

    \ \ \ \ \ \ \ \ \=lim_(h rarr 0) ( (sinx(cos h-1))/h+(sin hcosx) / h )

    \ \ \ \ \ \ \ \ \=lim_(h rarr 0) (sinx(cos h-1))/h+lim_(h rarr 0)(sin hcosx) / h

    \ \ \ \ \ \ \ \ \=(sinx)lim_(h rarr 0) (cos h-1)/h+(cosx)lim_(h rarr 0)(sin h) / h

    We know have to rely on some standard limits:

    lim_(h rarr 0)sin h/h =1 , and lim_(h rarr 0)(cos h-1)/h =0

    And so using these we have:

    f'(x)=0+(cosx)(1) =cosx

    Hence,

    d/dxsinx=cosx

    Steve M · 2 · Jun 12 2018

Questions

Differentiating Trigonometric Functions