Second Derivatives of Parametric Equations
AP Calculus BC · Flashcards · Recall
AP CALCULUS BC · FLASHCARDS
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QUESTION
State the second derivative formula in terms of
t
t
t
for
x
(
t
)
x(t)
x
(
t
)
and
y
(
t
)
y(t)
y
(
t
)
.
QUESTION
Identify the formula for
d
y
d
x
\frac{dy}{dx}
d
x
d
y
in terms of
x
(
t
)
x(t)
x
(
t
)
and
y
(
t
)
y(t)
y
(
t
)
.
QUESTION
State the formula for the second derivative in parametric form.
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ANSWER
d
2
y
d
x
2
=
d
d
t
(
d
y
d
x
)
d
x
d
t
\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}(\frac{dy}{dx})}{\frac{dx}{dt}}
d
x
2
d
2
y
=
d
t
d
x
d
t
d
(
d
x
d
y
)
. Apply the chain rule: differentiate
d
y
d
x
\frac{dy}{dx}
d
x
d
y
with respect to
t
t
t
, then divide by
d
x
d
t
\frac{dx}{dt}
d
t
d
x
.
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