(3) Find the derivative of tan x w. r. t. x.
Answers
Answered by
0
Answer:
First principle method,
f
′
(x)=lim
t→x
t−x
f(t)−f(x)
⇒f
′
(x)=lim
t→x
t−x
tan
t
−tan
x
=lim
t→x
t−x
1+tan
t
tan
x
tan
t
−tan
x
×(1+tan
t
tan
x
)
=lim
t→x
(
t−x
tan(
t
−
x
)
)×lim
t→x
(1+tan
t
tan
x
)
=(1+tan
t
tan
x
)lim
t→x
(
t
−
x
tan(
t
−
x
)
)
t
+
x
1
=(1+tan
2
x
)lim
(
t
−
x
)→0
(
t
−
x
tan(
t
−
x
)
)
t
+
x
1
=(sec
2
x
)(1)(
x
+
x
1
)
f
′
(x)=
2
x
1
sec
2
x
Answered by
2
First principle method,
f ′
lim
t→x
t−x
f(t)−f(x)
⇒
f ′
lim
t→x
t−x
tan
t
−tan
x
= lim
t→x
t−x
1+tan
t
tan
x
tan
t
−tan
x
× (1+
tan
t
tan
x
)
=
lim
t→x
(
t−x
tan(
t
−
x
)
)
×
lim
t→x
(1+
tan
t
tan
x
)
=
(1+
tan
t
tan
x
)lim
t→x
(
t
−
x
tan(
t
−
x
)
)
t
+
x
1
=
(1+
tan 2
x
)lim
(
t
−
x
)→0
(
t
−
x
tan(
t
−
x
)
)
t
+
x
1
= (sec
2
x
)(1) (
x
+
x
1
)
f ′
2
x
1
sec 2
x
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