The curve C is given by the equation x^4 + x^2y + y^2 = 13. Find the value of dy/dx at the point (-1,3). (A-level)

The most important thing to note about this question is that we must differentiate implicitly. We must differentiate each term with respect to x. Using our differentiation rules we can see that our first term x^4 when differentiated with respect to x becomes 4x^3. The second term is more complex as it requires both use of the product rule and implicit differentiation so that we can differentiate y with respect to x. Using the product rule formula, dy/dx = udv/dx + vdu/dx where u = x^2 and v = y. Differentiating each of these gives us du/dx = 2x using normal differentiation and dv/dx = 1dy/dx = dy/dx where we have differentiated implicitly in order to obtain y differentiated with respect to x. So, d/dx (x^2y) = x^2dy/dx +2xy. Finally to differentiate the final term of the left hand side we must use implicit differentiation. This gives us 2ydy/dx. The last term is 13 which is constant and so differentiates to 0. So now we have the expression 4x^3 + x^2dy/dx +2xy + 2y*dy/dx = 0.Now, we must rearrange to obtain an expression for dy/dx. This gives us dy/dx (x^2 + 2y) = -4x^3 - 2xy, which we rearrange by dividing both sides by (x^2 +2y), which leaves us with our required expression dy/dx = (-4x^3 - 2xy)/(x^2 + 2y). Then, substituting in the values given, x= -1 and y = 3 gives us our final answer of dy/dx = 10/7.