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Primitive function/Inverse function/Fact/Proof
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Primitive function/Inverse function/Fact
Proof
Differentiating, using
fact
and
fact
, yields
(
y
f
−
1
(
y
)
−
F
(
f
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1
(
y
)
)
)
′
=
f
−
1
(
y
)
+
y
1
f
′
(
f
−
1
(
y
)
)
−
f
(
f
−
1
(
y
)
)
1
f
′
(
f
−
1
(
y
)
)
=
f
−
1
(
y
)
.
{\displaystyle {}{\begin{aligned}{\left(yf^{-1}(y)-F{\left(f^{-1}(y)\right)}\right)}'&=f^{-1}(y)+y{\frac {1}{f'(f^{-1}(y))}}-f{\left(f^{-1}(y)\right)}{\frac {1}{f'{\left(f^{-1}(y)\right)}}}\\&=f^{-1}(y).\end{aligned}}}
To fact
