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Mathematics
Mathematics
Problem
If
x
=
f
(
t
)
x=f(t)
x
=
f
(
t
)
and
y
=
g
(
t
)
y=g(t)
y
=
g
(
t
)
are twice differentiable with
f
′
(
t
)
≠
0
f'(t)\neq0
f
′
(
t
)
=
0
, which of the following gives
d
2
y
d
x
2
\dfrac{d^{2}y}{dx^{2}}
d
x
2
d
2
y
in terms of
t
t
t
? a)
f
′
′
(
t
)
g
′
(
t
)
[
f
′
(
t
)
]
3
\displaystyle\frac{f''(t)\,g'(t)}{[f'(t)]^{3}}
[
f
′
(
t
)
]
3
f
′′
(
t
)
g
′
(
t
)
b)
g
′
′
(
t
)
[
f
′
(
t
)
]
2
\displaystyle\frac{g''(t)}{[f'(t)]^{2}}
[
f
′
(
t
)
]
2
g
′′
(
t
)
c)
f
′
(
t
)
g
′
′
(
t
)
−
g
′
(
t
)
f
′
′
(
t
)
[
f
′
(
t
)
]
3
\displaystyle\frac{f'(t)\,g''(t)-g'(t)\,f''(t)}{[f'(t)]^{3}}
[
f
′
(
t
)
]
3
f
′
(
t
)
g
′′
(
t
)
−
g
′
(
t
)
f
′′
(
t
)
d)
f
′
′
(
t
)
g
′
(
t
)
−
g
′
′
(
t
)
f
′
(
t
)
[
f
′
(
t
)
]
3
\displaystyle\frac{f''(t)\,g'(t)-g''(t)\,f'(t)}{[f'(t)]^{3}}
[
f
′
(
t
)
]
3
f
′′
(
t
)
g
′
(
t
)
−
g
′′
(
t
)
f
′
(
t
)
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