Find tho derivative
tant Wit x² + x)
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Answer:
can be derived using the inverse function theorem.
Take, for example, the function
y
=
f
(
x
)
=
arcsinh
x
(inverse hyperbolic sine). Together with the function
x
=
φ
(
y
)
=
sinh
y
they form a pair of mutually inverse funtions. Then the derivative of the inverse hyperbolic sine is given by
(
arcsinh
x
)
′
=
f
′
(
x
)
=
1
φ
′
(
y
)
=
1
(
sinh
y
)
′
=
1
cosh
y
=
1
√
1
+
sinh
2
y
=
1
√
1
+
sinh
2
(
arcsinh
x
)
=
1
√
1
+
x
2
.
Similarly, we can obtain the derivatives for the inverse hyperbolic cosine, tangent and cotangent functions.
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