A circle of radius 5 is defined implicitly by the relation .

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

Problem

A circle of radius 5 is defined implicitly by the relation

x^{2}+y^{2}=25

. (1) Differentiate both sides of the relation with respect to

x

and show that the resulting equation can be written in the form

2x+2y\dfrac{dy}{dx}=0

. Explain, in words, why an additional factor of

\dfrac{dy}{dx}

appears when differentiating the term

y^{2}

. Your explanation should make explicit reference to the fact that the coordinates

x

and

y

are not independent. (2) Using the result from part (a), describe how the sign and magnitude of

\dfrac{dy}{dx}

at a point

(x,y)

on the circle depend on the signs and magnitudes of

x

and

y

. Support your answer by interpreting

\dfrac{dy}{dx}

as the slope of the tangent line and by referring to geometric properties of the circle.