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Mathematics
Problem
If
y
=
f
(
g
(
x
)
)
y = f(g(x))
y
=
f
(
g
(
x
))
where
f
f
f
and
g
g
g
are twice differentiable functions, which of the following equals
d
2
y
d
x
2
\dfrac{d^{2}y}{dx^{2}}
d
x
2
d
2
y
? a)
f
′
′
(
g
(
x
)
)
g
′
′
(
x
)
+
f
′
(
g
(
x
)
)
[
g
′
(
x
)
]
2
f''(g(x))\,g''(x)+f'(g(x))\,[g'(x)]^{2}
f
′′
(
g
(
x
))
g
′′
(
x
)
+
f
′
(
g
(
x
))
[
g
′
(
x
)
]
2
b)
f
′
′
(
g
(
x
)
)
[
g
′
(
x
)
]
2
+
f
′
(
g
(
x
)
)
g
′
′
(
x
)
f''(g(x))\,[g'(x)]^{2}+f'(g(x))\,g''(x)
f
′′
(
g
(
x
))
[
g
′
(
x
)
]
2
+
f
′
(
g
(
x
))
g
′′
(
x
)
c)
f
′
′
(
g
(
x
)
)
[
g
′
(
x
)
]
2
−
f
′
(
g
(
x
)
)
g
′
′
(
x
)
f''(g(x))\,[g'(x)]^{2}-f'(g(x))\,g''(x)
f
′′
(
g
(
x
))
[
g
′
(
x
)
]
2
−
f
′
(
g
(
x
))
g
′′
(
x
)
d)
2
f
′
′
(
g
(
x
)
)
[
g
′
(
x
)
]
2
+
f
′
(
g
(
x
)
)
g
′
′
(
x
)
2\,f''(g(x))\,[g'(x)]^{2}+f'(g(x))\,g''(x)
2
f
′′
(
g
(
x
))
[
g
′
(
x
)
]
2
+
f
′
(
g
(
x
))
g
′′
(
x
)
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