Differentiating sin(x) from First Principles
Key Questions

Answer:
# d/dxsinx=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 h1)+sin hcosx ) / h # # \ \ \ \ \ \ \ \ \=lim_(h rarr 0) ( (sinx(cos h1))/h+(sin hcosx) / h )# # \ \ \ \ \ \ \ \ \=lim_(h rarr 0) (sinx(cos h1))/h+lim_(h rarr 0)(sin hcosx) / h# # \ \ \ \ \ \ \ \ \=(sinx)lim_(h rarr 0) (cos h1)/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 h1)/h =0 # And so using these we have:
# f'(x)=0+(cosx)(1) =cosx# Hence,
# d/dxsinx=cosx#
Questions
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Limits Involving Trigonometric Functions

Intuitive Approach to the derivative of y=sin(x)

Derivative Rules for y=cos(x) and y=tan(x)

Differentiating sin(x) from First Principles

Special Limits Involving sin(x), x, and tan(x)

Graphical Relationship Between sin(x), x, and tan(x), using Radian Measure

Derivatives of y=sec(x), y=cot(x), y= csc(x)

Differentiating Inverse Trigonometric Functions