Let and be differentiable functions of a parameter such that for all in their common…

MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

Let

x=f(t)

and

y=g(t)

be differentiable functions of a parameter

t

such that

f'(t)\neq 0

for all

t

in their common domain. (i) Starting from the expression

\dfrac{dy}{dx}=\dfrac{g'(t)}{f'(t)}

, derive an explicit formula for

\dfrac{d^{2}y}{dx^{2}}

in terms of

f'(t)

f''(t)

g'(t)

and

g''(t)

. (ii) Show each differentiation step separately and justify the transition from differentiation with respect to

t

to differentiation with respect to

x

. (iii) Verify that your final result reduces to

\dfrac{d^{2}y}{dx^{2}}=\dfrac{f'(t)g''(t)-g'(t)f''(t)}{\left[f'(t)\right]^{3}}