The curve defined implicitly by x^3+x y^3=12 passes through the point (2,1).\n(i) Find dy/dx at (2,1) using implicit differentiation.\n(ii) State whether the tangent at this point is horizontal, vertical, or neither.

dy/dx|_((2,1))=-1/3; the tangent is neither horizontal nor vertical.

The curve defined implicitly by $x^{3} + x y^{3} = 12$ passes through the point $(2, 1)$ .\n(i) Find $\frac{d y}{d x}$ at $(2, 1)$ using implicit differentiation.\n(ii) State whether the tangent at this point is horizontal, vertical, or neither.

hard 4 marks Class 12 · Mathematics · Continuity and Differentiability

Answer

\frac{d y}{d x}_{(2, 1)} = - \frac{1}{3}

; the tangent is neither horizontal nor vertical.

The correct answer is $\frac{d y}{d x}_{(2, 1)} = - \frac{1}{3}$ ; the tangent is neither horizontal nor vertical. — (ii) A horizontal tangent would require $\frac{d y}{d x} = 0$ ; vertical tangent would require denominator zero, i.e., $3 x y^{2} = 0$ . Neither occurs at $(2, 1)$ . Hence the tangent is neither horizontal nor vertical.

Step-by-step solution

(i) Differentiate implicitly: $3 x^{2} + y^{3} + 3 x y^{2} \frac{d y}{d x} = 0$ . Solve for $\frac{d y}{d x}$ : $\frac{d y}{d x} = \frac{- ( 3 x ^{2} + y ^{3} )}{3 x y ^{2}}$ .
Substitute $x = 2$ and $y = 1$ : numerator $- (3 \cdot 4 + 1) = - 13$ ; denominator $3 \cdot 2 \cdot 1^{2} = 6$ . Thus $\frac{d y}{d x} = \frac{- 13}{6}$ .
(ii) A horizontal tangent would require $\frac{d y}{d x} = 0$ ; vertical tangent would require denominator zero, i.e., $3 x y^{2} = 0$ . Neither occurs at $(2, 1)$ . Hence the tangent is neither horizontal nor vertical.

Common mistake: Forgetting to differentiate the product term correctly.

Concept tested: Implicit differentiation and slope of tangent

Published 19 September 2026 · How this solution was prepared and checked

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The curve defined implicitly by x3+xy3=12 passes through the point (2,1).\n(i) Find dxdy​ at (2,1) using implicit differentiation.\n(ii) State whether the tangent at this point is horizontal, vertical, or neither.

Step-by-step solution

The curve defined implicitly by $x^{3} + x y^{3} = 12$ passes through the point $(2, 1)$ .\n(i) Find $\frac{d y}{d x}$ at $(2, 1)$ using implicit differentiation.\n(ii) State whether the tangent at this point is horizontal, vertical, or neither.