How to differentiate x^2 + y^2 - 2x + 6y = 5
This is an implicit function, meaning the x and y terms are not able to be seperated onto different sides, i.e. y=f(x). Therefore, we must use implicit differentiation. Before we differentiate the equation given, recall that when differentiating y with respect to (w.r.t) x, we get an answer of dy/dx. This is because we have differentiated y normally w.r.t y first, and then multiplied it by dy/dx. For example, if we differentiate 2y w.r.t x, we get 2 dy/dx. Now, in order to differentiate the equation above, take it one term at a time, as follows:1. Differentiate x2 w.r.t x, which gives 2x2. Differentiate y2: differentiate y2 w.r.t y first, which gives 2y, then multiply by dy/dx, giving 2y dy/dx.3. Differentiate -2x => -24. Differentiate 6y => 6 dy/dx 5. (And on the right side) differentiate 5 => 0Then, put all the terms together, in the new differentiated equation:2x + 2y dy/dx - 2 + 6 dy/dx = 0 The final step is simply to rearrange the above equation to make dy/dx the subject of the formula. When done you should get:dy/dx = (2 - 2x) / (2y + 6)And there you have it!
