Find the equation for the tangent to the curve y^3 + x^3 + 3x^2 + 2y + 8 = 0 at the point (2,1)

As the tangent has the same gradient as the curve at the particular point, we need to find the gradient of the curve. To find the gradient we need to differentiate. Due to there being two variables in this equation it is difficult to differentiate. For this example, it is too difficult to rearrange to make y the subject which will make the equation harder to differentiate so we use implicit differentiation. When differentiating x terms its the same as normal differentiation (by bringing the power down to the front and then decreasing the power by 1). For example, 4x^2 would become (4)(2)x^(2-1) = 8x^1 which is just 8x. For y terms, we differentiate as if it is an x term but we multiply it with dy/dx at the end as we are differentiating with respect to x. So for example, y^4 would become (4)y^(4-1)dy/dx = = 4y^3(dy/dx) so for this question, when we differentiate we get (3)y^(3-1)(dy/dx) + (3)x^(3-1) + (3)(2)x^(2-1) +(2)(1)y^(1-1)(dy/dx) + 0 = 0 (2y is the same as 2y^1 and differentiating constants like 8 gives 0). When simplified, this becomes 3y^2(dy/dx) + 3x^2 + 6x + 2(dy/dx) = 0 (anything to the power of zero is 1). Now rearrange to find (dy/dx). 3y^2(dy/dx) + 2(dy/dx) = -3x^2 - 6x. We can now take a factor of (dy/dx) from the right hand side of the equation which gives (dy/dx)(3y^2 +2) = -3x^2 - 6x. Then by dividing both sides by (3y+2) we get the equation in terms of dy/dx which looks like dy/dx = (-3x^2 - 6x)/(3y^2 +2). To find the gradient at the point (2,1) sub the value for of 2 into x and 1 into y for the last equation. This will give us that dy/dx = -24/5. This is the gradient of the curve at that point and therefore is also the gradient of the tangent at the point. Then by using the equation y - y1 = m(x - x1) we can calculate the equation of the tangent. We just need to sub in the values into this equation to find the equation of the tangent, where y1 = 1, m = -24/5 and x = 2. Which gives us y - 1 = -24/5(x - 2). We can leave the answer like this as the question doesn't ask us to put the equation in a particular form. So our final answer is  y - 1 = -24/5(x - 2).