(General proof of higher-order derivative for a parametric curve) Let and be twice…

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

Problem

(General proof of higher-order derivative for a parametric curve) Let

x=f(t)

and

y=g(t)

be twice differentiable functions of the parameter

t

, with

f'(t)\neq 0

for all

t

in the interval of interest. (i) Starting from the definitions of

dx/dt

and

dy/dt

, derive an expression for

dy/dx

in terms of

t

. No result from textbooks may be quoted; all steps must be shown. (ii) Further differentiate your expression from part (i) with respect to

x

and obtain a formula for

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

expressed only in

f'(t),\,f''(t),\,g'(t)

and

g''(t)

. Simplify the final result as much as possible. (iii) State clearly the condition(s) under which the formula obtained in part (ii) is valid.