find the derivative of( sin x) using first principle
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Answer:
First principle of differentiation :
dx
dy
=lim
δx→0
δx
f(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(
2
C+D
)sin(
2
C−D
)
⇒f(x+δx)−f(x)=2cos(
2
x+δx+x
)sin(
2
δx
)
⇒
dx
dy
=lim
δx→0
δx
2cos(x+
2
δx
)sin(
2
δx
)
⇒
dx
dy
=lim
δx→0
cos(x+
2
δx
)
2
δx
sin(
2
δx
)
⇒
dx
d(sinx)
=cosx as lim
x→0
x
sinx
=1
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