find the differentiation of tan(π/2-x)
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Step-by-step explanation:
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Answer:
Knowing that for every function
h
(
x
)
that can be written as
f
(
g
(
x
)
)
,
h
′
(
x
)
=
f
′
(
g
(
x
)
)
⋅
g
′
(
x
)
In this case we have
f
(
x
)
=
tan
(
x
)
and
g
(
x
)
=
π
⋅
x
2
.
The derivative of
g
(
x
)
is easily computed, since the derivative of a first degree polynomial is the leading coefficient, so:
g
′
(
x
)
=
π
2
The derivative of
f
(
x
)
you can either check from a reference table (or remember it) since it shows up reasonably a lot, or use the quotient rule:
tan
(
x
)
=
sin
(
x
)
cos
(
x
)
→
tan
′
(
x
)
=
cos
2
(
x
)
+
sin
2
(
x
)
cos
2
(
x
)
Using the identity
cos
2
(
x
)
+
sin
2
(
x
)
=
1
, we get that:
tan
′
(
x
)
=
1
cos
2
(
x
)
=
sec
2
(
x
)
So the derivative of
tan
(
π
⋅
x
2
)
=
π
2
⋅
sec
2
(
π
⋅
x
2
)
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