Use implicit differentiation to determine the gradient of curves whose equations are given in implicit form.

Consider the curve defined by \(x^3 + 2xy^2 - y^3 = 7\). (a) Use implicit differentiation to determine an expression for \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). (2 marks) (b) Determine the gradient of the tangent to the curve at the point \((2, 1)\). (1 mark) (c) Determine the equation of the normal to the curve at the point \((2, 1)\). Express your answer in the form \(ax + by + c = 0\), where \(a\), \(b\), and \(c\) are integers. (1 mark)

Use implicit differentiation to find the gradient of the curve x² + xy + y² = 7 at the point (2, 1).

Consider the curve defined by \(x^3 + 2xy^2 - y^3 = 7\). (a) Use implicit differentiation to determine an expression for \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). (2 marks) (b) Use your result from part (a) to determine the gradient of the tangent to the curve at the point \((2, 1)\). (1 mark) (c) Determine the equation of the tangent to the curve at the point \((2, 1)\). Express your answer in the form \(y = mx + c\). (1 mark)

Consider the curve defined by the equation \(x^3 + 2xy^2 - y^3 = 16\). (a) Use implicit differentiation to determine an expression for \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). (2 marks) (b) Show that the point \(A(2, 2)\) lies on the curve. (1 mark) (c) Determine the gradient of the tangent to the curve at \(A\). (1 mark) (d) Hence determine the equation of the normal to the curve at \(A\), giving your answer in the form \(ax + by + c = 0\) where \(a\), \(b\), and \(c\) are integers. (1 mark)

Find the gradient of the curve x² + xy + y² = 7 at the point (2, 1).

Use implicit differentiation to find the gradient of the curve at the point (2, 1).