The one-parameter family of curves is defined by where and is a positive constant.

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

The one-parameter family of curves is defined by

x^{n}+y^{n}=a^{n}

where

n>1

and

a

is a positive constant. (i) Derive a general formula for

\dfrac{dy}{dx}

in terms of

x,\,y

and

n

.},{ (ii) Hence obtain a general expression for

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

for the family. (iii) Show that, at points on the curve where

x=y

, the second derivative simplifies to an expression involving only

x

and

n

. State this simplified form.