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NCERT Class XI Mathematics - Limits and Derivatives - Solutions

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Question : 42
Total: 72
Find the derivative of cos x from first principle.
Solution:  
Let f(x) = cos x
We have,
d
dx
 (f (x)) =
lim
h→0
f(x+h)−f(x)
h
 =
lim
h→0
cos(x+h)−cosx
h
=
lim
h→0
[
−2sin(
x+h+x
2
).sin(
x+h−x
2
)
h
]

=
lim
h→0
[
−2sin(
2x+h
2
).sin(
h
2
)
h
]
=
lim
h→0
[
−2sin(x+
h
2
).sin
h
2
h
]
=
lim
h→0
[
−sin(x+
h
2
).sin(
h
2
)
h
2
]
=
lim
h→0
[−sin(x+
h
2
).
sin(
h
2
)
h
2
]
= - [
lim
h→0
sin(x+
h
2
)][
lim
h→0
sin
h
2
h
2
] = - sin x
∴
d
dx
 (cos x) = - sin x.
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