Given the equation 3x^2 + 4xy - y^2 + 12 = 0. Solve for dy/dx in terms of x and y.

Here the key is to remember to differentiate with both x and y with respect to x, where the differential of y is dy/dx. Consider the first term, 3x² : This differentiates to 6x. This is done by multiplying the coefficient and the power and then subtracting one from the power ie. (3x2)x^2-1 = 6x. Consider the second term, 4xy. To differentiate this, we must use the chain rule, this means differentiating first x and then y. First with rx, here you consider y to be a constant (ie. just a number we are multiplying the coefficient by), so we differentiate to get, since x has a power of 1, 4y. Then differentiate with y with respect to x, remembering the differential of y is dy/dx, then with y having a power of 1, we get (4x1)x(dy/dx). We then add the differentials together, according to the chain rule to get 4y + 4x(dy/dx). Then consider the third term, -y² : This differentiates to -2y(dy/dx). As we did with 3x² , we multiply the coefficient by the power and subtract one from the power, however here we remember that y differentiates to dy/dx. So, (-1x2)y^2-1(dy/dx) = -2y(dy/dx). The final term is a constant, and therefore differentiates to 0 leaving us with 6x + 4y +4x(dy/dx) -2y(dy/dx) = 0. To find dy/dx, you then rearrange: 6x + 4y = 2y(dy/dx) - 4x (dy/dx) Then taking dy/dx out: 6x + 4y = (dy/dx)(2y - 4x) Divide through by 2y - 4x: dy/dx = (6x + 4y)/(2y - 4x) = (3x + 2y)/(y - 2x)