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If and are inverse functions of each other
MATHEMATICS · Single correct · ISC · Mathematics · Calculus · Calculus
Problem
If
y
=
f
(
x
)
y=f(x)
y
=
f
(
x
)
and
x
=
g
(
y
)
x=g(y)
x
=
g
(
y
)
are inverse functions of each other, then (1)
g
′
′
(
y
)
=
1
f
′
(
x
)
g''(y)=\frac{1}{f'(x)}
g
′′
(
y
)
=
f
′
(
x
)
1
(2)
g
′
′
(
y
)
=
−
f
′
′
(
x
)
(
f
′
(
x
)
)
3
g''(y)=-\frac{f''(x)}{(f'(x))^3}
g
′′
(
y
)
=
−
(
f
′
(
x
)
)
3
f
′′
(
x
)
(3)
g
′
′
(
y
)
=
−
f
′
(
x
)
(
f
′
′
(
x
)
)
3
g''(y)=-\frac{f'(x)}{(f''(x))^3}
g
′′
(
y
)
=
−
(
f
′′
(
x
)
)
3
f
′
(
x
)
(4) None of these
Related theory
Derivatives of Inverse Functions
Advanced Differentiation
Implicit and Parametric Differentiation
Derivatives of standard functions via chain rule
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