An implicit curve is defined by the equation .

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

Problem

An implicit curve is defined by the equation

x^{2}+y^{2}+xy=7

. (1) Calculate

\dfrac{dy}{dx}

in terms of

x

and

y

. Show all algebraic steps that lead to your final simplified expression. (2) Evaluate

\dfrac{dy}{dx}

at the point

(2,1)

and interpret the numerical value you obtain. (3) Using your result from part (b), write the equation of the tangent line to the curve at

(2,1)

. State explicitly whether this tangent line rises or falls as

x

increases and justify your conclusion.