Derivative f’ of the function f(x)=sinx is: f’(x) = cos x for any value of x.

Derivative of sin x

Derivative f of the function f(x)=sinx is: x],+[,f(x)=cosx

Proof/Demonstration

sin(x+h)sinxh=sin(x)cos(h)+cos(x)sin(h)sinxhsin(x+h)sinxh=sinhh×cosx+sinx×coshhsinx×1hsin(x+h)sinxh=sinhh×cosx+sinx×cosh1h

We have

cosh1h=(cosh1)(cosh+1)h(cosh+1)=cos2h1h(cosh+1)=sin2hh(cosh+1)=sinhh×sinhcosh+1

Then

limh0cosh1h=0

because

limh0sinhh=1 This equality has been proved in /limits/article/proof-of-limit-of-sin-x-x-1-as-x-approaches-0

Now

limh0sin(x+h)sinxh=limh0sinhh×cosx+sinx×limh0cosh1h=1×cosx+sinx×0

We conclude:

limh0sin(x+h)sinxh=cosx