Question

Find the derivative of sinx by first principle.

Answer 1

First principle of differentiation :

dxdy =lim δx→0δxf(x+δx)−f(x

Here f(x)=sinx

⇒f(x+δx)=sin(x+δx)

⇒f(x+δx)−f(x)=sin(x+δx)−sinx

We know that sinC−sinD=2cos( 2C+D )sin( 2C−D )

⇒f(x+δx)−f(x)=2cos( 2x+δx+x )sin( 2δx )

⇒ dxdy =lim δx→0δx2cos(x+ 2δx )sin(2δx )

⇒ dxdy =lim δx→0cos(x+ 2δ ) 2δxsin( 2δx )

⇒dxd(sinx) =cosx as lim x→0xsinx=1