Find the derivative of f(x)=tan(x/x+1) by first principle.
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Explanation:
From the first principle of derivatives,
f
′
(x)=
h→0
lim
h
f(x+h)−f(x)
=
h→0
lim
h
tan(x+h)−tanx
=
h→0
lim
h
cos(x+h)
sin(x+h)
−
cosx
sinx
=
h→0
lim
hcosxcos(x+h)
cosxsin(x+h)−sinxcos(x+h)
=
h→0
lim
hcosxcos(x+h)
2
sin(2x+h)+sinh
−
2
sin(2x+h)−sinh
=
h→0
lim
hcosxcos(x+h)
sinh
=
h→0
lim
h
sinh
×
h→0
lim
cosxcos(x+h)
1
=1×
cosx×cosx
1
=
cos
2
x
1
f
′
(x)=sec
2
x
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